Question

What polynomial has roots of −4, −1, and 6? x3 − 9x2 − 22x + 24 x3 − x2 − 26x − 24 x3 + x2 − 26x + 24 x3 + 9x2 + 14x − 24

Answer 1

Answer:  The answer is (b) $x^3-x^2-26x-24.$

Step-by-step explanation:  We are given four polynomials and we are to check which one of them has roots -4, -1 and 6. Obviously, if putting these three values of 'x' in a polynomials yields 0, then that particular value will be a root of that polynomial.

Let us denote the polynomials as follows -

$P(x)=x^3-9x^2-22x+24,\\\\Q(x)=x^3-x^2-26x-24,\\\\R(x)=x^3+x^2-26x+24\\\\\textup{and}\\\\S(x)=x^3+9x^2+14x-24.$

Let us check for x = -1 first. So, substituting x = -1 in all the four polynomils, we get

$P(-1)=36\neq 0,~~Q(-1)=0,~~R(x)=50\neq 0~~\textup{and}~~S(x)=-30\neq 0.$

Therefore, only possibility is Q(x).

If we put x = -4 and x = 6 in Q(x), we find that

$Q(-4)=0~~\textup{and}~~Q(6)=0.$

Thus, the correct option is (b) $x^3-x^2-26x-24.$