Question

Find all the zeros of the polynomial 2x4 − 5x3 − 11x2 + 20x + 12 if it is given that two of its
zeroes are 2 and -2.

Answer 1

We are given that two of its zeroes are 2 and -2.

This means two factors are (x-2) and (x+2).

Therefore we divide by (x²-4) to find other factors.

$\sf{2x^4 - 5x^3 - 11x^2 + 20x + 12=(x^2-4)(2x^2-5x-3)}$

Therefore it will have two roots at:

• $\sf{(x^2-4)=0}$
• $\sf{(2x^2-5x-3)=0}$

So it means:

• $\sf(x+2)(x-2)=0}$
• $\sf{(2x+1)(x-3)=0}$

Therefore all zeros are at:

• $\sf{x=2}$
• $\sf{x=-2}$
• $\sf{x=-\dfrac{1}{2} }$
• $\sf{x=3}$

For your information.

We know two factors because of the factor theorem.

• Reason: The converse of the factor theorem is true.

The zero of the polynomial is where each factor becomes 0.

• Reason: When a number is multiplied by 0, it is always 0.