Question

For question 12, use the graph to guess one zero for the polynomial whose graph is shown. Then, factor completely.

12) f(x) = x³ - 6x² + 6x + 8​

Answer 1

x³ - 6·x² + 6·x + 8 = (x - 4) × (x - (1 + √3)) × (x + 1 + √3)

Explanation:

From the given graph of the polynomial, x³ - 6·x² + 6·x + 8, we have that one of the zeros of the polynomial, which is one point the graph cuts the x-axis is given as follows;

At a zero, we have, x = 4

∴ x - 4 = 0 is a factor of the polynomial

Dividing the given polynomial by (x - 4), gives;

x² - 2·x - 2

(x³ - 6·x² + 6·x + 8)÷(x - 4)

x³ -4·x²

-2·x² + 6·x + 8

-2·x² + 8·x

-2·x + 8

-2·x + 8

0

∴ (x³ - 6·x² + 6·x + 8)÷(x - 4) = x² - 2·x - 2

∴ (x - 4) × (x² - 2·x - 2) = x³ - 6·x² + 6·x + 8

Factorizing x² - 2·x - 2, gives;

x = (2 ± √((-2)² - 4×1×(-2)))/(2 × 1) = (2 ± 2·√(3)/(2) = (1 ± √(3)

Therefore, x = 1 + √3, (x - (1 + √3)) = 0 or x = 1 - √3, (x - (1 - √3)) = 0 which gives the factors, after factorizing completely as follows;

x³ - 6·x² + 6·x + 8 = (x - 4) × (x - (1 + √3)) × (x + 1 + √3)