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

\(x=4\) 
 5 x squared minus 8 x plus 4.
Substitute 4 for x.5 times 4 squared minus 8 times 4 plus 4.
Simplify the exponents.5 times 16 minus 8 times 4 plus 4.
Multiply.80 minus 32 plus 4.
Simplify.52.
\(x=-2\) 
 5 x squared minus 8 x plus 4.
Substitute negative 2 for x.5 times negative 2 squared minus 8 times negative 2 plus 4.
Simplify the exponents.5 times 4 minus 8 times negative 2 plus 4.
Multiply.20 plus 16 plus 4.
Simplify.40.
\(x=0\) 
 5 x squared minus 8 x plus 4.
Substitute 0 for x.5 times 0 squared minus 8 times 0 plus 4.
Simplify the exponents.5 times 0 minus 8 times 0 plus 4.
Multiply.0 plus 0 plus 4.
Simplify.4.

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

 6 x squared plus 15 x y.
Substitute x equals 4 and y equals 6.6 times 4 squared plus 15 times 4 times 6.
Simplify.6 times 16 plus 15 times 4 times 6.
Simplify.96 plus 360.
Simplify.456.
 The cost of producing the box is $456.

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