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26 Polynomials

Topics Covered:[1]

Identifying the Degree and Leading Coefficient of Polynomials

We have learned that a term is a constant or the product of a constant and one or more variables.

A monomial is an algebraic expression with one term. When it is of the form axm, where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomial in one variable are 8y2, -3x. Monomials can also have more than one variable such as −4a2b3c2.

binomial—A polynomial with exactly two terms is called a binomial.

trinomial—A polynomial with exactly three terms is called a trinomial.

The table below has some examples of polynomials.

Type Ex. 1 Ex. 2 Ex. 3 Ex. 4
Polynomial y + 1 8y2 -14x 4a2 − 7ab + 2b2 4x4 + x3 + 8x2 − 9x+1
Monomial 14 8y2 −9x3y5 −13a3b2c
Binomial a + 7b 4x2 − y2 y2 − 16 3p3q − 9p2q
Trinomial x2 − 7x + 12 9m2 + 2mn − 8n2 6k4 − k3 + 8k z4 + 3z2 − 1

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.
A monomial that has no variable, just a constant, is a special case. The degree of a constant is then 0. In the table above, the example of monomial “14” would have a degree of 0.

 Degree

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let’s start by looking at a monomial. The monomial 8ab2 has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn’t have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, 1 + 2, is 3, so the degree is 3.

Try it!

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

a. 7y2 − 5y + 3  b. −2a4b2  c. 3x5 − 4x3 − 6x2 + x − 8  d. 2y−8xy3  e. 15

Solution (click to reveal)
Polynomial Number of terms Type Degree of terms Degree of polynomial
7y2 − 5y + 3 3 Trinomial 2, 1, 0 2
−2a4b2 1 Monomial 4, 2 6
3x5 − 4x3 − 6x2 + x − 8 5 Polynomial 5, 3, 2, 1, 0 5
2y − 8xy3 2 Binomial 1, 4 4
15 1 Monomial 0 0

In a polynomial, which is a sum of or difference of terms, each term consists of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as the 384 in  384x, is known as a coefficient.

Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product aixi, such as 384xw, is a term of a polynomial. If a term does not contain a variable, it is called a constant.

The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

\displaystyle\boldsymbol{\underbrace{\color{myorange1}{a}\color{black}{_nX}\color{myblue1}{^n}\color{black}}_{\textcolor{myblue1}{\textbf{Leading Term}}}{+\ldots+a_2X^2+a_1X+a_0}}

  • In the formula above, \displaystyle\boldsymbol{\color{myorange1}{a}} represents the LEADING COEFFICIENT.
  • Where \displaystyle\boldsymbol{x\color{myblue1}{^n}}, the \displaystyle\boldsymbol{\color{myblue1}{^n}} is the DEGREE

Polynomials

A polynomial is an expression that can be written in the form anxn + … + a2x2 + a1x + a0

Each real number ai is called a coefficient. The number a0that is not multiplied by a variable is called a constant. Each product aixi is a term of a polynomial. The highest power of the variable that occurs in the polynomial is called the degree of a polynomial. The leading term is the term with the highest power, and its coefficient is called the leading coefficient.

Given a polynomial expression, identify the degree and leading coefficient.

  1. Find the highest power of x to determine the degree.
  2. Identify the term containing the highest power of x to find the leading term.
  3. Identify the coefficient of the leading term.

Try it!

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

  1. 3 + 2x2 − 4x3     b. 5t5 − 2t3 + 7     c. 6p − p3 − 2
Solution (click to reveal)
  1. The highest power of x is 3, so the degree is 3. The leading term is the term containing that degree, −4x3. The leading coefficient is the coefficient of that term, −4.
  2. The highest power of t is 5, so the degree is 5. The leading term is the term containing that degree, 5t5. The leading coefficient is the coefficient of that term, 5.
  3. The highest power of p is 3, so the degree is 3. The leading term is the term containing that degree, −p3, The leading coefficient is the coefficient of that term, −1.

Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, 5x2 and −2x2 are like terms, and can be added to get 3x2, but 3x and 3x2 are not like terms, and therefore cannot be added.

Try it!

Add or subtract: a. 25y2 + 15y2       b.  16pq3 − (−7pq3).

 

Solution A (click to reveal)
a.

Steps Algebraic
Problem 25y2 + 15y2
Combine like terms.    40y2

 

Solution B (click to reveal)

b.

Steps Algebraic
Problem   16pq3 − (−7pq3)
Combine like terms. 23pq3

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent.

Commutative Property

The Commutative Property allows us to rearrange the terms to put like terms together.

  1. Combine like terms.
  2. Simplify and write in standard form.

Try it!

Find the sum:  (7y2 − 2y + 9) + (4y2 − 8y − 7).

Solution (click to reveal)
Steps Algebraic
Identify like terms. (7y2 − 2y + 9) + (4y2 − 8y − 7)
Rewrite without the parentheses, rearranging to get the like terms together.      7y2 + 4y2 − 2y − 8y + 9 − 7
Combine like terms.            11y2 − 10y + 2

Try it!

Find the sum.

(12x2 + 9x − 21) + (4x3 + 8x2 − 5x + 20)

Solution (click to reveal)
Steps Algebraic
Combine like terms.            4x3 + (12x2 + 8x2) + (9x − 5x) + (−21 + 20)     
Simplify. 4x3 + 20x2 + 4x − 1

Analysis

We can check our answers to these types of problems using a graphing calculator. To check, graph the problem as given along with the simplified answer. The two graphs should be equivalent. Be sure to use the same window to compare the graphs. Using different windows can make the expressions seem equivalent when they are not.

Tip!

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Try it!

Find the difference:  (9w2 − 7w + 5) − (2w2 − 4).

Solution (click to reveal)
Steps Algebraic
Problem    (9w2 − 7w + 5) − (2w2 − 4)
Distribute and identify like terms.  9w2 − 7w + 5 – 2w2 + 4
Rearrange the terms.  9w2 − 2w2 − 7w + 5 + 4
Combine like terms. 7w2 − 7w + 9

Try it!

Find the difference.

(7x4 − x2 + 6x + 1) − (5x3 − 2x2 + 3x + 2)

Solution (click to reveal)
Steps Algebraic
Combine like terms.      7x4 − 5x3 + (−x2 + 2x2) + (6x − 3x) + (1 − 2)
Simplify.        7x4 − 5x3 + x2 + 3x − 1

Analysis

Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.

Multiplying Polynomials

Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.

Try it!

Multiply:  a. (3x2)(−4x3)   b. (\frac{5}{6}x3y)(12xy2).

Solution A (click to reveal)

a.

Steps Algebraic
Problem  (3x2)(−4x3)
Use the Commutative Property to rearrange the terms. 3 · (−4) · x2 · x3
Multiply. -12x5

 

Solution B (click to reveal)

b. 

Steps Algebraic
Problem   \boldsymbol{\displaystyle\left(\frac{5}{6}x^3y\right)\left(12xy^2\right)}
Use the Commutative Property to rearrange the terms. \boldsymbol{\displaystyle\frac{5}{6}\cdot 12\cdot x^3\cdot x\cdot y\cdot y^2}
Multiply. 10x4y3

Multiplying a polynomial by a monomial is really just applying the Distributive Property.

Notice that before combining like terms, you have four terms. You multiply the two terms of the first binomial by the two terms of the second binomial—four multiplications. See the table below for a demonstration.

Steps Algebraic
Distribute p  

\begin{aligned} \boldsymbol{\downarrow\!\color{myred1}{\xleftarrow{\phantom{000}}}}&}\\ \boldsymbol{\left(x+3\right)\color{myred1}{p}}& \\ \boldsymbol{\uparrow\!\color{myred1}{\xleftarrow{\phantom{00000}}}}&}\\ \end{aligned}
We distributed the p to get: \displaystyle\boldsymbol{x\color{myred1}{p}\color{black}{+3}\color{myred1}{p}}
What if we have (x + 7) instead of p?  

\begin{aligned} \boldsymbol{\downarrow\color{myred1}{\xleftarrow{\phantom{00000}}\phantom{00}}}&}\\ \displaystyle\boldsymbol{\left(x+3\right)\color{myred1}{\left(x+7\right)}}& \\ \boldsymbol{\uparrow\color{myred1}{\xleftarrow{\phantom{00000000}}\phantom{00}}}&}\\ \end{aligned}
Distribute (x + 7).  

\begin{aligned} \boldsymbol{ {\color{myred1}{\xrightarrow{\phantom{00}}}{\color{black}{|}} \phantom{000000}} {\color{myred1}{\xrightarrow{\phantom{00}}}}{\color{black}{|}}\phantom{0000}}\\ \displaystyle\boldsymbol{ x\color{myred1}{\left(x+7\right)}\color{black}{+3}\color{myred1}{\left(x+7\right)}} \\ \boldsymbol{ {\color{myred1}{\xrightarrow{\phantom{00000}}}{\color{black}{|}}\phantom{000}} {\color{myred1}{\xrightarrow{\phantom{00000}}}}{\color{black}{|}}\phantom{00}}\\ \end{aligned}
Distribute again. \displaystyle\boldsymbol{x^2+7x+3x+21}
Combine like terms. \displaystyle\boldsymbol{x^2+10x+21}

Try it!

Multiply:  a. −2y(4y2 + 3y − 5)  b. 3x3y(x2 − 8xy + y2)

Solution A (click to reveal)

a.

Steps Algebraic
Example \displaystyle\boldsymbol{-2y\left(4y^2+3y-5\right)}
Distribute -2y. \displaystyle\boldsymbol{{\color{myred1}{-2y}}\cdot 4y^2+\left({\color{myred1}{-2y}}\right)\cdot 3y-\left({\color{myred1}{-2y}}\right)\cdot 5}
Multiply. \displaystyle\boldsymbol{-8y^3-6y^2+10y}

 

Solution B (click to reveal)

b.

Steps Algebraic
Example   3x3y(x2 − 8xy + y2)
Distribute  3x3y \displaystyle\boldsymbol{\left({\color{myred1}{ 3x^3y}}\right) \cdot x^2 + \left({\color{myred1}{3x^3y}}\right) \cdot\left(−8xy\right) + \left({\color{myred1}{ 3x^3y}}\right)\cdot y^2}
Multiply.             3x5y − 24x4y2 + 3x3y3

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. We can distribute the 2 in 2(x+7) to obtain the equivalent expression 2x+14. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

Given the multiplication of two polynomials, use the distributive property to simplify the expression.

  1. Multiply each term of the first polynomial by each term of the second.
  2. Combine like terms.
  3. Simplify.

Try it!

Find the product.

(2x + 1) (3x2 − x + 4)

Solution (click to reveal)
Steps Algebraic
Use the distributive property. 2x(3x2 − x + 4) + (3x2 − x + 4)
Multiply.  (6x3 − 2x2 + 8x) + (3x2 − x + 4)
Combine like terms.   6x3 + (−2x2 + 3x2) + (8x − x) + 4
Simplify.           6x3 + x2 + 7x + 4
Analysis

We can use a table to keep track of our work. Write one polynomial across the top and the other down the side, The first cell will be blank. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.

— 3x2  -x +4
2x 6x3 -2x2 8x
+1 3x2 -x +4

Try it!

Multiply (b + 3)(2b2 – 5b + 8) using a. the Distributive Property and b. the Vertical Method.

Solution A (click to reveal)

a.

Steps Algebraic
Problem \displaystyle\boldsymbol{\left(b+3\right)\left(2b^2-5b+8\right)}
Distribute (2b2 – 5b + 8). \displaystyle\boldsymbol{b{\color{myred1}\left(2b^2-5b+8\right)}+3{\color{myred1}\left(2b^2-5b+8\right)}}
Multiply. \displaystyle\boldsymbol{2b^3-5b^2+8b+6b^2-15b+24}
Combine like terms. \displaystyle\boldsymbol{2b^3+b^2-7b+24}

 

Solution B (click to reveal)

b. It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.

Steps Algebraic
Multiply (2b2 − 5b + 8) by 3.                                                

    \[ \begin{array}{r} \phantom{+}\boldsymbol{2b^2 - 5b + 8} \\ \times \phantom{+}\boldsymbol{b + 3} \\ \hline \phantom{+}\boldsymbol{6b^2 - 15b + 24} \\ \end{array} \]
Multiply (2b2 − 5b + 8) by b. \displaystyle\boldsymbol{2b^3-5b^2+8b}
Combine like terms. \displaystyle\boldsymbol{2b^3+b^2-7b+24}

Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial.

\displaystyle\boldsymbol{\left({\color{myblue1}{ax}}+{\color{myorange1}{b}}\right)\left({\color{myblue1}{cx}}+{\color{myorange1}{d}}\right)={\color{myblue1}{acx^2}}+{\color{myblue1}{a}}{\color{myorange1}{d}}{\color{myblue1}{x}}+{\color{myorange1}{b}}{\color{myblue1}{cx}}+{\color{myorange1}{bd}}}}

Using the equation above we can break down FOIL:

  • FIRST TERMS are indicated with \boldsymbol{\color{myblue1}{blue}}: The ax in the first expression and the cx in the 2nd expression.
  • OUTER TERMS are indicated with: The \boldsymbol{\color{myblue1}{blue}} ax in the first expression and the \boldsymbol{\color{myorange1}{orange}} d in the 2nd expression.
  • INNER TERMS are indicated with: The \boldsymbol{\color{myblue1}{blue}} c in the first expression and the \boldsymbol{\color{myorange1}{orange}} d in the 2nd expression.
  • LAST TERMS  are indicated with \boldsymbol{\color{myorange1}{orange}}: The b in the first expression and the d in the 2nd expression.

The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.

Given two binomials, use FOIL to simplify the expression in the table below:

Steps Algebraic
Step 1. Multiply the First terms. \displaystyle\boldsymbol{\left(\color{myred1}{a}\color{black}{+}b\right)\left(\color{myred1}{c}\color{black}{+}d\right)}
Step 2. Multiply the Outer terms. \displaystyle\boldsymbol{\left(\color{myred1}{a}\color{black}{+}b\right)\left(c+\color{myred1}{d}\color{black}{\right)}}
Step 3. Multiply the Inner terms. \displaystyle\boldsymbol{\left(a+\color{myred1}{b}\color{black}{\right)}\left(\color{myred1}{c}\color{black}{+}d\right)}
Step 4. Multiply the Last terms. \displaystyle\boldsymbol{\left(a+\color{myred1}{b}\color{black}{\right)}\left(c+\color{myred1}{d}\color{black}{\right)}}
Step 5. Combine like terms, when possible Say it as you multiply! FOIL First Outer Inner Last

 

Try it!

Use FOIL to find the product:  (2x – 18)(3x + 3)

Solution (click to reveal)
Steps Algebraic
Find the product of the first terms.

    \begin{gather*}\boldsymbol{\left({\color{myred1}2x}-18\right)\left({\color{myred1}3x}+3\right)}\\\textbf{so}\\\boldsymbol{{\color{myred1}2x\cdot 3x} = 6x^2} \end{gather*}}\]
Find the product of the outer terms.

    \begin{gather*} \boldsymbol{\left({\color{myred1}2x}-18\right)\left(3x+{\color{myred1}3}\right)}\\ \textbf{so}\\ \boldsymbol{{\color{myred1}2x\cdot 3} = 6x} \end{gather*}
Find the product of the inner terms.

    \begin{gather*} \boldsymbol{\left(2x{\color{myred1}-18}\right)\left({\color{myred1}3x}+3\right)}\\ \textbf{so}\\ \boldsymbol{{\color{myred1}-18\cdot 3x}=-54x} \end{gather*}
Find the product of the last terms.

    \begin{gather*} \mathbf{\left(2x{\color{myred1}-18}\right)\left(3x+{\color{myred1}3}\right)} \\ \textbf{so}\\ \mathbf{{\color{myred1}-18\cdot 3}=-54} \end{gather*}
Add the products. 6x2 + 6x − 54x − 54
Combine like terms. 6x2 + (6x − 54x) − 54
Simplify. 6x2 − 48x − 54

The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.

    \[ \begin{array}{rl} \begin{array}{r} \phantom{+}\boldsymbol{23} \\ \boldsymbol{\times}\phantom{+}\boldsymbol{46} \\ \hline \phantom{+}\boldsymbol{138} \\ \boldsymbol{92}\phantom{0} \\ \hline \boldsymbol{1058} \\ \end{array} & \begin{array}{l}\\\\ \textbf{ Start by multiplying 23 by 6 to get 138. } \\ \textbf{Next, multiply 23 by 4, lining up the partial product in the correct columns.} \\ \textbf{Last you add the partial products.} \end{array} \end{array} \]

Try it!

Multiply using the Vertical Method:

(3y − 1) (2y − 6)

Solution (click to reveal)

It does not matter which binomial goes on the top.

Steps Algebraic: FOIL method Algebraic: Partial products
Use either Foil method or Partial products

 

 

    \[ \begin{array}{r} \\\\ \phantom{+}\boldsymbol{\left(3y-1\right)\left(2y-6\right)} \\ \phantom{+}\boldsymbol{\color{myblue1}{6y^2-2y}\color{myred1}{-18y+6}} \\ \hline \phantom{+}\boldsymbol{6y^2-20y+6}\\ \end{array} \]

    \[ \begin{array}{l} \phantom{++}\boldsymbol{3y-1} \\ \boldsymbol{\times}\phantom{+}\boldsymbol{2y-6} \\ \hline \phantom{++}\boldsymbol{\color{myred1}{-18y+6}} \\ \boldsymbol{\color{myblue1}{6y^2-2y}}\phantom{0} \\ \hline \boldsymbol{6y^2-20y+6} \end{array} \]

Notice the partial products are the same as the terms in the FOIL method.

Sum or Difference (aka Difference of Squares or Conjugates)

Another special product is called the difference of squares, or Sum and Difference of Two Terms.

A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference is called a conjugate pair and is of the form (a − b), (a + b).

In the first table below, 3 conjugate pair examples are listed. Notice how each pair has one sum and one difference. The first cell is left blank.

— Ex. 1, binomial 1  Ex. 1, binomial 2 Ex. 2, binomial 1  Ex. 2, binomial 2  Ex. 3, binomial 1  Ex. 3, binomial 2
Binomial \displaystyle\boldsymbol{\left(x\color{myred1}{-}\color{black}{9}\right)} \displaystyle\boldsymbol{\left(x\color{myred1}{+}\color{black}{9}\right)} \displaystyle\boldsymbol{\left(y\color{myred1}{-}\color{black}{8}\right)} \displaystyle\boldsymbol{\left(y\color{myred1}{+}\color{black}{8}\right)} \displaystyle\boldsymbol{\left(2x\color{myred1}{-}\color{black}{5}\right)} \displaystyle\boldsymbol{\left(2x\color{myred1}{+}\color{black}{5}\right)}
Label Difference Sum Difference Sum Difference Sum

The next table includes 2 examples of a conjugate pair that demonstrates the result of finding the product.

Example 1 
 Example 2
\displaystyle\boldsymbol{\left(x+\color{myred1}{9}\color{black}{\right)}\left(x-\color{myred1}{9}\color{black}{\right)}} \displaystyle\boldsymbol{\left(2x-\color{myred1}{5}\color{black}{\right)}\left(2x+\color{myred1}{5}\color{black}{\right)}}
\displaystyle\boldsymbol{\color{myblue1}{x^2}\color{black}{-9x}+9x-\color{myred1}{81}} \displaystyle\boldsymbol{\color{myblue1}{4x^2}\color{black}{+10x}-10x-\color{myred1}{25}}
\displaystyle\boldsymbol{\color{myblue1}{x^2}\color{black}{-}\color{myred1}{81}} \displaystyle\boldsymbol{\color{myblue1}{4x^2}\color{black}{-}\color{myred1}{25}}

Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

Difference of Squares or Sum and Difference (Conjugates)

When a binomial is multiplied by a binomial with the same terms separated by the opposite sign, the result is the square of the first term minus the square of the last term.

\displaystyle\boldsymbol{\left(a-b\right)\left(a+b\right) =a^2-b^2\quad\quad\quad\underbrace{\left(a-b\right)\left(a+b\right)}_{conjugates} = \underset {squares}{a^2} \quad \quad \overset {difference}{-}\quad\quad\underset { squares}{b^2}}

Try it!

Multiply:     (x − 8) (x + 8).

Solution (click to reveal)

First, recognize this as a product of conjugates. The binomials have the same first terms, and the same last terms, and one binomial is a sum and the other is a difference.

Steps Algebraic
It fits the pattern. \boldsymbol{\binom{{\color{myred1}a-b}}{x-8}\binom{{\color{myred1}a+b}}{x+8}}
Square the first term, x. The conjugate form of a2-b2 will be displayed above the example. The unfilled as yet 2nd term will be displayed as an empty box. 

    \begin{gather*} \boldsymbol{{\color{myred1}a^2 - b^2}}\\ \boldsymbol{x^2- \square} \end{gather*}
Square the last term, 8.

    \begin{gather*} \boldsymbol{{\color{myred1}a^2 - b^2}}\\ \boldsymbol{x^2- 8^2} \end{gather*}
The product is a difference of squares.

    \begin{gather*} \boldsymbol{{\color{myred1}a^2 - b^2}}\\ \boldsymbol{x^2 - 64} \end{gather*}

Binomial (Perfect) Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier and faster. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.

(x + 5)2 = x2 + 10x + 25

(x − 3)2 = x2 − 6x + 9 

(4x − 1)2 = 16x2 − 8x + 1

Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.

Perfect Square Trinomials

When a binomial is squared, the result is the first term squared added to double the product of both terms and the last term squared.

(x + a)2 = (x + a)(x + a) = x2 + 2ax + a2

Or

(a + b)2 = a2 + 2ab + b2 Representation
Algebraic form (a + b)2 = a2 + 2ab + b2
Description (binomial)2 equal sign (first term)2 plus sign 2(product of terms) plus sign (last term)2

 

(a – b)2 = a2 – 2ab + b2 Representation
Algebraic form (a – b)2 = a2 – 2ab + b2
Description (binomial)2 equal sign (first term)2 minus sign 2(product of terms plus sign (last term)2

Given a binomial, square it using the formula for perfect square trinomials.

  1. Square the first term of the binomial.
  2. Square the last term of the binomial.
  3. For the middle term of the trinomial, double the product of the two terms.
  4. Add and simplify.

Try it!

Multiply:  a. (x + 5)2  b. (2x − 3y)2.

Solution (click to reveal)
Steps Algebraic
Example \displaystyle\boldsymbol{\binom{{\color{myred1}a+b}}{x+5}^2}
Square the first term.

The perfect square form of a2+2ab+b2 will be displayed above the example. The unfilled as yet 2nd and 3rd terms will be displayed as an empty box. 

    \begin{gather*} \boldsymbol{{\color{myred1}a^2 +2ab+ b^2}}\\ \boldsymbol{x^2+ \square + \square} \end{gather*}
Square the last term.

    \begin{gather*}\boldsymbol{{\color{myred1}a^2 +2ab+ b^2}}\\ \boldsymbol{x^2+ \square + 5^2} \end{gather*}
Double their product. \displaystyle\boldsymbol{{\color{myred1}a^2+2\cdot a\cdot b+b^2}}
\displaystyle\boldsymbol{x^2+2\cdot x\cdot 5+5^2}
Simplify. x2 + 10x + 25

 

Solution B (click to reveal)

b.

Steps Algebraic
Example \displaystyle\boldsymbol{\binom{{\color{myred1}a-b}}{2x-3y}^2}
Use the pattern. \displaystyle\boldsymbol{{\color{myred1}a^2-2\cdot a\cdot b+b^2}}
\displaystyle\boldsymbol{\left(2x\right)^2-2\cdot 2x\cdot 3y+\left(3y\right)^2}
Simplify. 4x2 – 12xy + 9y2

Try it!

Expand (4x − 1)2.

Solution (click to reveal)

16x2 − 8x + 1

Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example in the table below:

(a + 2b) (4a − b − c)

Steps Algebraic
Use the distributive property.             a(4a − b − c) + 2b(4a − b − c)
Multiply.     4a2 − ab − ac + 8ab − 2b2 − 2bc
Combine like terms.      4a2 + (−ab + 8ab) − ac − 2b2 – 2bc
Simplify.   4a2 + 7ab − ac − 2bc − 2b2

Try it!

Find the sum:  (u2 − 6uv + 5v2) + (3u2 + 2uv)
Solution (click to reveal)
Steps Algebraic
Use the distributive property.                                                           u2 − 6uv + 5v2 + 3u2 + 2uv
Rearrange the terms to put like terms together. u2 + 3u2 − 6uv + 2uv + 5v2
Combine like terms.   4u2 − 4uv + 5v2

Try it!

Multiply  (x + 4) (3x − 2y + 5).

Solution (click to reveal)

Follow the same steps that we used to multiply polynomials containing only one variable.

Steps Algebraic
Use the distributive property.                                                       x(3x − 2y + 5) + 4(3x − 2y + 5)
Multiply.      3x2 − 2xy + 5x + 12x − 8y + 20
Combine like terms.     3x2 − 2xy + (5x + 12x) − 8y + 20
Simplify.      3x2 − 2xy + 17x − 8y + 20

Try it!

Multiply  (3x − 1) (2x + 7y − 9).

Solution (click to reveal)

6x2 + 21xy − 29x − 7y + 9

Access these online resources for additional instruction and practice with polynomials.

    Key Concepts

    • Polynomial
      • A polynomial is a sum of terms each consisting of a variable raised to a non-negative integer power.
      • The degree is the highest power of the variable that occurs in the polynomial.
      • The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term.
    • Add and Subtract Polynomials We can add and subtract polynomials by combining like terms.
    • Multiply Polynomials
      • To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products.
      • FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials.
      • Perfect square trinomials and difference of squares are special products.
      • Follow the same rules to work with polynomials containing several variables.
    Binomial Squares Sum or Difference (Product of Conjugates)
    (a + b)2 = a2 + 2ab + b2 (a − b)(a + b) = a2 − b2
    (a − b)2 = a2 − 2ab + b2 —
    Squaring a binomial Multiplying conjugates
    Product is a trinomial Product is a binomial.
    Inner and outer terms with FOIL are the same. Inner and outer terms with FOIL are opposites.
    Middle term is double the product of the terms There is no middle term.
    1. Derived from Openstax Intermediate Algebra; Access for free at https://openstax.org/books/intermediate-algebra-2e/pages/1-introduction. Elementary Algebra; Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction. College Algebra 2e with Corequisite support; Access for free at https://openstax.org/books/college-algebra-corequisite-support-2e/pages/1-introduction-to-prerequisites ↵
    definition

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