Use synthetic division to determine if the given value for k is a zero of this polynomial. If not, determine p(k).

Question

Use synthetic division to determine if the given value for  is a zero of this polynomial. If not, determine .

p(x)=2x³−3x²−11x−20; k=4

is k a zero of this polynomial?

Answer 1

When the remainder of the division equals zero, it is proof that the divisor is a factor of the dividend.

Given p(x) = 2x³ - 3x² - 11x - 20 and k = 4:

4 | 2 -3 -11 -20 given

4 | 2 -3 -11 -20 lower the 2

2

4 | 2 -3 -11 -20 multiply 4 * 2 = 8; -3 + 8 = 5

8

2 5

4 | 2 -3 -11 -20 multiply 4 * 5 = 20; -11 + 20 = 9

8 20

2 5 9

4 | 2 -3 -11 -20 multiply 4 * 9 = 36; -20 + 36 = 16

8 20 36

2 5 9 16

p(k) = p(4) = 16, according to the Remainder Theorem

Since the remainder is not equal zero, then k = 4 is not a zero of the polynomial.

Answer 2

4| 2 -3 -11 -20

8 20 36

2 5 9 16

P(4) = 16. By the remainder theorem, 4 is not a zero because the remainder is a nonzero.

If you use Desmos and type in the polynomial function, then the x-intercept is around (3.709, 0). In this case, k = 4 is not one of the zeros.