Question

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

Answer 1

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(-1) = 2(-1)^4+(-1)^3-3(-1)^2+5(-1)`
`f(-1)= 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(x) = 2x^4+1x^3-3x^2+5x+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.