Question

Use synthetic division to determine whether the number k is an upper or lower bound (as specified) for the real zeros of the function f. k = -1; f(x) = 4x^3 - 2x^2 + 2x + 4; Lower bound? yes/no?

Answer 1

Yes, k is lower bound for the real zeros of the function.

Explanation:

Lower Bound Theorem:

If you divide a polynomial function f(x) by (x - k), where k < 0, using synthetic division and this yields alternating signs, then k is a lower bound to the real roots of the equation f(x) = 0.

Given;

4x³ - 2x² + 2x + 4

k = -1

synthetic division;

-1 ] 4 -2 2 4

↓ -4 6 -8

---------------------------------------------

4 -6 8 -4

(the result has alternating signs, hence k is lower bound)

Thus, k is lower bound for the real zeros of the function.