Question

Which of the following are roots of the polynomial function F(x) = x^3 - 5x^2 - 13x - 7

A. 7
B. 3 + Square root of 2
C. 1 + square root of 3
D. -1
E. 1 - square root of 3
F. 3 - square root of 2

Answer 1

The roots are: 1, -7 and -7

Explanation:

When we put x = -1 in F(x) it is equal to 0

Thus, (x + 1) is factor of given function.

When we divide F(x) by (x + 1) it gives the value (x² - 6x - 7)

Now, factorizing x² - 6x - 7 by middle term splitting.

x² - 6x - 7 = x² - 7x + x - 7

⇒ x(x - 7) + 1(x - 7)

⇒ (x + 1) (x - 7)

Thus, x³ - 5x² - 13x - 7 = (x + 1)(x + 1)(x - 7)

Thus, Roots of the given polynomial function is x = -1, -1, 7

Hence, option A and D are correct.

Answer 2

D and A

Explanation:

If you don't know how to solve a cubic algebraically (and who really does) then the best way is something like Desmos.

I can't copy the graph from this computer, but the answer is two roots are

x = - 1 and the other one is x = 7

So you would write it as (x + 1)(x + 1)(x - 7) in factor form.

Answer: D and A

I got the graph finally. It is loaded below.