Add and Subtract Polynomials

5.1 Add and Subtract Polynomials

Topics covered in this section are:

  1. Determine the degree of polynomials
  2. Add and subtract polynomials
  3. Evaluate a polynomial function for a given value
  4. Add and subtract polynomial functions

5.1.1 Determine the Degree 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 $ax^{m}$, 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. Monomials can also have more than one variable such as and $−4a^{2}b^{3}c^{2}$.

MONOMIAL

monomial is an algebraic expression with one term.

A monomial in one variable is a term of the form $ax^{m}$, where $a$ is a constant and $m$ is a whole number.

A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.

POLYNOMIALS

polynomial—A monomial, or two or more algebraic terms combined by addition or subtraction is a polynomial.

monomial—A polynomial with exactly one term is called a monomial.

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

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

Here are some examples of polynomials.

Polynomial$y+1$$4a^{2}-7ab+2b^{2}$$4x^{4}+x^{3}+8x^{2}-9x+1$
Mononomial$14$$8y^{2}$$-9x^{3}y^{5}$$-13a^{3}b^{2}c$
Binomial$a+7b$$4x^{2}-y^{2}$$y^{2}-16$$3p^{3}q-9p^{2}q$
Trinomial$x^{2}-7x+12$$9m^{2}+2mn-8n^{2}$$6x^{4}-k^{3}+8k$$z^{4}+3z^{2}-1$

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomialbinomial, and trinomialwhen referring to these special polynomials and just call all the rest polynomials.

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 $0$.

DEGREE OF A POLYNOMIAL

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 see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Let’s start by looking at a monomial. The monomial $8ab^{2}$ 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 $3$.

The polynomial is 8 a b squared. The exponents of the variables are 1 and 2 so the degree of the monomial is 1 plus 2 which equals 3.

Here are some additional examples.

Monomial examples: 14 has degree 0, 8 a b squared has degree 3, negative 9 x cubed y to the fifth power has degree 8, negative 13 a has degree 1. Binomial examples: The terms in h plus 7 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 7 b squared minus 3 b have degree 2 and 1 so the degree of the whole polynomial is 2. The terms in z squared y squared minus 25 have degree 4 and 0 so the degree of the whole polynomial is 4. The terms in 4 n cubed minus 8 n squared have degree 3 and 2 so the degree of the whole polynomial is 3. Trinomial examples: The terms in x squared minus 12 x plus 27 have degree 2, 1 and 0 so the degree of the whole polynomial is 2. The terms in 9 a squared plus 6 a b plus b squared have degree 2, 2, and 2 so the degree of the whole polynomial is 2. The terms in 6 m to the fourth power minus m cubed n squared plus 8 m n to the fifth power have degree 4, 5, and 6 so the degree of the whole polynomial is 6. The terms in z to the fourth power plus 3 z squared minus 1 have degree 4, 2, and 0 so the degree of the whole polynomial is 4. Polynomial examples: The terms in y minus 1 have degree 1 and 0 so the degree of the whole polynomial is 1. The terms in 3 y squared minus 2 y minus 5 have degree 2, 1, 0 so the degree of the whole polynomial is 2. The terms in 4 x to the fourth power plus x cubed plus eight x squared minus 9 x plus 1 have degree 4, 3, 2, 1, and 0 so the degree of the whole polynomial is 4.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

Example 1

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

  • $7y^{2}-5y+3$
  • $-2a^{4}b^{2}$
  • $3x^{5}-4x^{3}-6x^{2}+x-8$
  • $2y-8xy^{3}$
  • $15$
Solution
PolynomialNumber of termsTypeDegree of termsDegree of polynomial
$7y^{2}-5y+3$$3$Trinomial$2, 1, 0$$2$
$-2a^{4}b^{2}$$1$Monomial$6$$6$
$3x^{5}-4x^{3}-6x^{2}+x-8$$5$Polynomial$5, 3, 2, 1, 0$$5$
$2y-8xy^{3}#$2$Binomial$1, 4$$4$
$15$$1$Monomial$0$$0$

5.1.2 Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

Example 2

Add or subtract:

  • $25y^{2}+15y^{2}$
  • $16pq^{3}-(-7pq^{3})$
Solution
$25y^{2}+15y^{2}$
Combine like terms.$40y^{2}$
$16pq^{3}-(-7pq^{3})$
Combine like terms.$23pq^{3}$

Remember that like terms must have the same variables with the same exponents.

Example 3

Simplify:

  • $a^{2}+7b^{2}-6a^{2}$
  • $u^{2}v+5u^{2}-3v^{2}$
Solution
$a^{2}+7b^{2}-6a^{2}$
$-5a^{2}+7b^{2}$
$u^{2}v+5u^{2}-3v^{2}$
There are no like terms to combine. In this case, the polynomial is unchanged.$u^{2}v+5u^{2}-3v^{2}$

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

Example 4

Find the sum: $(7y^{2}-2y+9)+(4y^{2}-8y-7)$.

Solution
Identify like terms.$\left(\underline{\underline{7y^{2}}}- \underline{2y} +9 \right)+ \left(\underline{\underline{4y^{2}}}-\underline{8y}-7 \right)$
Rewrite without the parentheses, rearranging to get the like terms together.$\underline{\underline{7y^{2}+4y^{2}}} – \underline{2y-8y}+9-7$
Combine like terms.$11y^{2}-10y+2$

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

Example 5

Find the difference: $(9w^{2}-7w+5)-(2w^{2}-4)$.

Solution
$(9w^{2}-7w+5)-(2w^{2}-4)$
Distribute and identify like terms.$\underline{9w^{2}}-7w+5-\underline{2w^{2}}+4$
Rearrange the terms.$\underline{9w^{2}-2w^{2}}-7w+5+4$
Combine like terms.$7w^{2}-7w+9$

To subtract $a$ from $b$, we write it as $b−a$, placing the $b$ first.

Example 6

Subtract $(p^{2}+10pq-2q^{2})$ from $(p^{2}+q^{2})$.

Solution
$(p^{2}+q^{2})-(p^{2}+10pq-2q^{2})$
Distribute.$p^{2}+q^{2}-p^{2}-10pq+2q^{2}$
Rearrange the terms, to put like terms together.$p^{2}-p^{2}-10pq+q^{2}+2q^{2}$
Combine like terms.$-10pq+3q^{2}$

Example 7

Find the sum: $(u^{2}-6uv+5v^{2})+(3u^{2}+2uv)$.

Solution
$(u^{2}-6uv+5v^{2})+(3u^{2}+2uv)$
Distribute.$u^{2}-6uv+5v^{2}+3u^{2}+2uv$
Rearrange the terms to put liker terms together.$u^{2}+3u^{2}-6uv+2uv+5v^{2}$
Combine like terms.$4u^{2}-4uv+5v^{2}$

When we add and subtract more than two polynomials, the process is the same.

Example 8

Simplify: $(a^{3}-a^{2}b)-(ab^{2}+b^{3})+(a^{2}b+ab^{2})$.

Solution
$(a^{3}-a^{2}b)-(ab^{2}+b^{3})+(a^{2}b+ab^{2})$
Distribute.$a^{3}-a^{2}b-ab^{2}-b^{3}+a^{2}b+ab^{2}$
Rearrange to get the like terms together.$a^{3}-a^{2}b+a^{2}b-ab^{2}-+ab^{2}-b^{3}$
Combine like terms.$a^{3}-b^{3}$

5.1.3 Evaluate a Polynomial Function for a Given Value

polynomial function is a function defined by a polynomial. For example, $f(x)=x^{2}+5x+6$ and $g(x)=3x−4$ are polynomial functions, because $x^{2}+5x+6$ and $3x−4$ are polynomials.

POLYNOMIAL FUNCTION

polynomial function is a function whose range values are defined by a polynomial.

In Chapter 3, where we first introduced functions, we learned that evaluating a function means to find the value of $f(x)$ for a given value of $x$. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example 9

For the function $f(x)=5x^{2}-8x+4$ find:

  • $f(4)$
  • $f(-2)$
  • $f(0)$
Solution

Part 1

$f(x)=5x^{2}-8x+4$
To find $f(4)$, substitute $\textcolor{red}{4}$ for $x$.$f(\textcolor{red}{4})=5(\textcolor{red}{4})^{2}-8(\textcolor{red}{4})+4$
Simplify the exponents.$f(4)=5\cdot 16-8(4)+4$
Multiply.$f(4)=80-32+4$
Simplify.$f(4)=52$

Part 2

$f(x)=5x^{2}-8x+4$
To find $f(-2)$, substitute $\textcolor{red}{-2}$ for $x$.$f(\textcolor{red}{-2})=5(\textcolor{red}{-2})^{2}-8(\textcolor{red}{-2})+4$
Simplify the exponents.$f(-2)=5\cdot 4-8(-2)+4$
Multiply.$f(-2)=20+16+4$
Simplify.$f(-2)=40$

Part 3

$f(x)=5x^{2}-8x+4$
To find $f(-2)$, substitute $\textcolor{red}{0}$ for $x$.$f(\textcolor{red}{0})=5(\textcolor{red}{0})^{2}-8(\textcolor{red}{0})+4$
Simplify the exponents.$f(0)=5\cdot 0-8(0)+4$
Multiply.$f(0)=0+0+4$
Simplify.$f(0)=4$

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from $250$ ft.

Example 10

The polynomial function $h(t)=−16t^{2}+250$ gives the height of a ball $t$ seconds after it is dropped from a $250$-foot tall building. Find the height after $t=2$ seconds.

Solution
$h(t)=-16t^{2}+250$
To find $h(2)$, substitute $t=2$$h(2)=-16(2)^{2}+250$
Simplify.$h(2)=-16 \cdot 4 +250$
Simplify.$h(2)=-64+250$
Simplify.$h(2)=186$
After $2$ seconds, the height of the ball is $186$ feet.

5.1.4 Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

ADDITION AND SUBTRACTION OF POLYNOMIAL FUNCTIONS

For functions $f(x)$ and $g(x)$,

$(f+g)(x)=f(x)+g(x)$
$(f-g)(x)=f(x)-g(x)$

Example 11

For functions $f(x)=3x^{2}-5x+7$ and $g(x)=x^{2}-4x-3$, find:

  • $(f+g)(x)$
  • $(f+g)(3)$
  • $(f-g)(x)$
  • $(f-g)(-2)$
Solution

Part 1

$(f+g)(x)=f(x)+g(x)$
Substitute $f(x)=\textcolor{red}{3x^{2}-5x+7}$ and $g(x)=\textcolor{blue}{x^{2}-4x-3}$.$(f+g)(x)=(\textcolor{red}{3x^{2}-5x+7})+(\textcolor{blue}{x^{2}-4x-3})$
Rewrite without the parentheses.$(f+g)(x)=3x^{2}-5x+7+x^{2}-4x-3$
Put like terms together.$(f+g)(x)=3x^{2}+x^{2}-5x-4x+7-3$
Combine like terms.$(f+g)(x)=4x^{2}-9x+4$

Part 2

In part 1 we found $(f+g)(x)$ and now are asked to find $(f+g)(3)$.

$(f+g)(x)=4x^{2}-9x+4$
To find $(f+g)(3)$, substitute $x=3$.$(f+g)(3)=4(3)^{2}-9(3)+4$
$(f+g)(3)=4(9)-9(3)+4$
$(f+g)(3)=36-27+4$
$(f+g)(3)=13$

Notice that we could have found $(f+g)(3)$ by first finding the values of $f(3)$ and $g(3)$ separately and then adding the results.

Find $f(3)$.$\begin{align*} f(x)&=3x^{2}-5x+7 \\ f(\textcolor{red}{3})&=3(\textcolor{red}{3})^{2}-5(\textcolor{red}{3})+7 \\ f(3)&=3(9)-5(3)+7 \\ f(3)&=19 \end{align*}$
Find $g(3)$.$\begin{align*} g(x)&=x^{2}-4x-3 \\ g(\textcolor{red}{3})&=(\textcolor{red}{3})^{2}-4(\textcolor{red}{3})-3 \\ g(3)&=9-4(3)-3 \\ g(3)&=-6 \end{align*}$
Find $(f+g)(3)$$(f+g)(x)=f(x)+g(x)$
$(f+g)(3)=f(3)+g(3)$
Substitute $f(3)=\textcolor{red}{19}$ and $g(3)=\textcolor{blue}{-6}$.$(f+g)(3)=\textcolor{red}{19}+(\textcolor{blue}{-6})$
$(f+g)(3)=13$

Part 3

$(f-g)(x)=f(x)-g(x)$
Substitute $f(x)=\textcolor{red}{3x^{2}-5x+7}$ and $g(x)=\textcolor{blue}{x^{2}-4x-3}$.$(f-g)(x)=(\textcolor{red}{3x^{2}-5x+7})-(\textcolor{blue}{x^{2}-4x-3})$
Rewrite without the parentheses.$(f-g)(x)=3x^{2}-5x+7-x^{2}+4x+3$
Put like terms together.$(f-g)(x)=3x^{2}-x^{2}-5x+4x+7+3$
Combine like terms.$(f-g)(x)=2x^{2}-x+10$

Part 4

$(f-g)(x)=2x^{2}-x+10$
To find $(f-g)(-2)$, substitute $x=-2$.$(f-g)(-2)=2(\textcolor{red}{-2})^{2}-(\textcolor{red}{-2})+10$
$(f-g)(-2)=2\cdot4-(-2)+10$
$(f-g)(-2)=8+2+10$
$(f-g)(-2)=20$
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