Question

Wich of the following is the complete list of roots for the polynomial function f(x) = (x²+2x-15) (x²+8x+17)?
-5, 3
-5, 3, -4 + i, -4-i
-5, 3, -4+1, 4+i
-4+i,-4-i

Answer 1

Answer:

$-5, \quad 3, \quad -4+i,\quad -4-i$

Step-by-step explanation:

Roots occur when f(x) = 0

$\implies (x^2+2x-15)(x^2+8x+17)=0$

First trinomial

$\implies (x^2+2x-15)=0$

$\implies x^2+5x-3x-15=0$

$\implies x(x+5)-3(x+5)=0$

$\implies (x-3)(x+5)=0$

Therefore, roots are 3 and -5

Second trinomial

The second trinomial cannot be factored, so solve using the quadratic formula:

$x=\dfrac{-b \pm \sqrt{b^2-4ac} }{2a}\quad\textsf{when }\:ax^2+bx+c=0$

Given:  $(x^2+8x+17)=0$

Therefore:  a = 1, b = 8 and c = 17

$\implies x=\dfrac{-8 \pm \sqrt{8^2-4(1)(17)}}{2(1)}$

$\implies x=\dfrac{-8 \pm \sqrt{-4}}{2}$

$\implies x=\dfrac{-8 \pm \sqrt{4}\sqrt{-1}}{2}$

$\implies x=\dfrac{-8 \pm 2i}{2}$

$\implies x=-4 \pm i$

So the roots of the second trinomial are -4 + i and -4 - i

Therefore, the complete list of roots for the given polynomial function are:

$-5, \quad 3, \quad -4+i,\quad -4-i$