8.2: Evaluating a Polynomial for Given Values of the Variables

Learning Outcomes

Evaluate a polynomial for a given value

Key words

Evaluate: find the value of

Evaluating a Polynomial

Previously we evaluated expressions by “plugging in” numbers for variables. Since polynomials are expressions, we’ll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify. To evaluate an expression for a value of the variable, we substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.

example

Evaluate $3{x}^{2}-9x+7$ when

1. $x=3$
2. $x=-1$

Solution

1. $x=3$
	$3{x}^{2}-9x+7$
Substitute $3$ for $x$	$3{\left(3\right)}^{2}-9\left(3\right)+7$
Simplify the expression with the exponent.	$3\cdot 9 - 9\left(3\right)+7$
Multiply.	$27 - 27+7$
Simplify.	$7$

2. $x=-1$
	$3{x}^{2}-9x+7$
Substitute $−1$ for $x$	$3{\left(-1\right)}^{2}-9\left(-1\right)+7$
Simplify the expression with the exponent.	$3\cdot 1 - 9\left(-1\right)+7$
Multiply.	$3+9+7$
Simplify.	$19$

Example

Evaluate $3x^{2}-2x+1$ for $x=-1$ .

Solution

Substitute $-1$ for each $x$ in the polynomial:

$3x^{2}-2x+1=3\left(-1\right)^{2}-2\left(-1\right)+1$

Following the order of operations, evaluate exponents first:

$3\left(1\right)-2\left(-1\right)+1$

Multiply from left to right:

$3+2+1$

Add:

$3+2+1$

$6$

Answer

$3x^{2}-2x+1=6$ , for $x=-1$

Example

Evaluate $\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p$ for $p = 3$ .

Solution

Substitute 3 for each p in the polynomial.

$\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=\displaystyle -\frac{2}{3}\left(3\right)^{4}+2\left(3\right)^{3}-(3)$

Following the order of operations, evaluate exponents first:

$\displaystyle -\frac{2}{3}\left(81\right)+2\left(27\right)-3$

Multiply:

$\large -54 + 54 – 3$

Add and then subtract:

$-3$

Answer

$\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=-3$ , for $p = 3$

try it

The following video presents examples of evaluating a polynomial for a given value.

The following video provides another example of how to evaluate a polynomial for a negative number.

Example

Evaluate $3x^2-6x+4$ when $x=-\frac{2}{3}$

Solution

$\begin{equation}\begin{aligned}&3x^2-6x+4 \;\;\;\;\;\;\;\;\;\;\text{substitute }x\text{ with }-\frac{2}{3}\\ &= 3\left ( -\frac{2}{3}\right )^2-6\left ( -\frac{2}{3}\right )+4 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{evaluate exponent}\\ &= \frac{3}{1}\cdot \frac{4}{9}-\frac{6}{1}\cdot -\frac{2}{3}+4 \;\;\;\;\;\;\;\;\;\;\text{multiply}\\ &=\frac{4}{3}+4+4 \;\;\;\;\;\;\;\;\;\;\text{simplify}\\ &=\frac{4}{3}+\frac{16}{1} \;\;\;\;\;\;\;\;\;\;\text{add using a common denominator}\\ &=\frac{4+16\cdot 3}{3} \;\;\;\;\;\;\;\;\;\;\text{multiply}\\ &= \frac{4+48}{3}\;\;\;\;\;\;\;\;\;\;\text{simplify}\\ &= \frac{52}{3}\end{aligned}\end{equation}$

Try It

Evaluate $x^3+4x^2$ when $x=-\frac{1}{2}$

Show Answer

$\frac{7}{8}$

Try It

Evaluate $x^2+4x-2$ when $x=-\frac{3}{8}$

Show Answer

$-\frac{215}{64}$

Chapter 8: Polynomials