Mathematics » Polynomials II » Add and Subtract Polynomials

# Evaluating a Polynomial For a Given Value

## Evaluating a Polynomial For a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

## Example

Evaluate $$5{x}^{2}-8x+4$$ when

1. $$x=4$$
2. $$x=-2$$
3. $$x=0$$

### Solution

 $$x=4$$      Simplify the exponents.  Multiply.  Simplify.  $$x=-2$$      Simplify the exponents.  Multiply.  Simplify.  $$x=0$$      Simplify the exponents.  Multiply.  Simplify.  ## Example

The polynomial $$-16{t}^{2}+250$$ 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

$$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}-16{t}^{2}+250\hfill \\ \text{Substitute}\phantom{\rule{0.2em}{0ex}}t=2.\hfill & & & \phantom{\rule{4em}{0ex}}-16{\left(2\right)}^{2}+250\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}-16·4+250\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}-64+250\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}186\hfill \\ & & & \phantom{\rule{4em}{0ex}}\text{After 2 seconds the height of the ball is 186 feet.}\hfill \end{array}$$

## Example

The polynomial $$6{x}^{2}+15xy$$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $$x=4$$ feet and $$y=6$$ feet.

### Solution      Simplify.  Simplify.  Simplify.  The cost of producing the box is \$456.