# How do you evaluate f(x)=2x^4+x^3-3x^2+5x at x=-1 using direct substitution and synthetic division?

Sep 9, 2016

Substitute -1 into the original equation for direct substitution and use the remainder in synthetic division/

#### Explanation:

To find the answer using direct substitution, just plug in -1 for x.

$f \left(- 1\right) = 2 {\left(- 1\right)}^{4} + {\left(- 1\right)}^{3} - 3 {\left(- 1\right)}^{2} + 5 \left(- 1\right)$
$f \left(- 1\right) = 2 - 1 - 3 - 5 = - 7$

To evaluate the polynomial with synthetic division, use all the coefficients as the dividend and the -1 as the divisor. Be careful when examining the coefficients. The constant term at the end of the equation is 0, so a zero must be used as the last term in the dividend.

( Ignore all the dots in the synthetic division problem. I needed to use them to line up the numbers because spaces do not show up in the answer.)

$f \left(x\right) = 2 {x}^{4} + 1 {x}^{3} - 3 {x}^{2} + 5 x + 0$

-1 ]..2....1....-3....5....0
............-2.....1....2...-7
....------------------------
.......2...-1...-2....7....-7

To do synthetic division, pull down the first coefficient 2 below the line. Multiply the divisor -1 by this 2. The product -2 is then written under the next coefficient 1. Add the 1 and the -2. Write the sum -1 under the line. Multiply the divisor -1 by the sum -1. Write the product 1 under the next coefficient, -3. Add the -3 and the 1. Write the sum -2 under the line. Repeat multiplying and adding. The final number written under the line is -7.

-7 is the remainder, and is also the answer to f(-1). BTW, using the remainder of synthetic division to find evaluate a polynomial is known as the Remainder Theorem.