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 [latex]3{x}^{2}-9x+7[/latex] when
1. [latex]x=3[/latex]
2. [latex]x=-1[/latex]
Solution
| 1. [latex]x=3[/latex] | |
| [latex]3{x}^{2}-9x+7[/latex] | |
| Substitute [latex]3[/latex] for [latex]x[/latex] | [latex]3{\left(3\right)}^{2}-9\left(3\right)+7[/latex] |
| Simplify the expression with the exponent. | [latex]3\cdot 9 - 9\left(3\right)+7[/latex] |
| Multiply. | [latex]27 - 27+7[/latex] |
| Simplify. | [latex]7[/latex] |
| 2. [latex]x=-1[/latex] | |
| [latex]3{x}^{2}-9x+7[/latex] | |
| Substitute [latex]−1[/latex] for [latex]x[/latex] | [latex]3{\left(-1\right)}^{2}-9\left(-1\right)+7[/latex] |
| Simplify the expression with the exponent. | [latex]3\cdot 1 - 9\left(-1\right)+7[/latex] |
| Multiply. | [latex]3+9+7[/latex] |
| Simplify. | [latex]19[/latex] |
Example
Evaluate [latex]3x^{2}-2x+1[/latex] for [latex]x=-1[/latex].
Solution
Substitute [latex]-1[/latex] for each [latex]x[/latex] in the polynomial:
[latex]3x^{2}-2x+1=3\left(-1\right)^{2}-2\left(-1\right)+1[/latex]
Following the order of operations, evaluate exponents first:
[latex]3\left(1\right)-2\left(-1\right)+1[/latex]
Multiply from left to right:
[latex]3+2+1[/latex]
Add:
[latex]3+2+1[/latex]
[latex]6[/latex]
Answer
[latex]3x^{2}-2x+1=6[/latex], for [latex]x=-1[/latex]
Example
Evaluate [latex] \displaystyle -\frac{2}{3}p^{4}+2p^{3}-p[/latex] for [latex]p = 3[/latex].
Solution
Substitute 3 for each p in the polynomial.
[latex]\displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=\displaystyle -\frac{2}{3}\left(3\right)^{4}+2\left(3\right)^{3}-(3)[/latex]
Following the order of operations, evaluate exponents first:
[latex] \displaystyle -\frac{2}{3}\left(81\right)+2\left(27\right)-3[/latex]
Multiply:
[latex]\large -54 + 54 – 3[/latex]
Add and then subtract:
[latex]-3[/latex]
Answer
[latex] \displaystyle -\frac{2}{3}p^{4}+2p^{3}-p=-3[/latex], for [latex]p = 3[/latex]
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 [latex]3x^2-6x+4[/latex] when [latex]x=-\frac{2}{3}[/latex]
Solution
[latex]\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}[/latex]
Try It
Evaluate [latex]x^3+4x^2[/latex] when [latex]x=-\frac{1}{2}[/latex]
Try It
Evaluate [latex]x^2+4x-2[/latex] when [latex]x=-\frac{3}{8}[/latex]
