Question

Which of the following is a polynomial with roots 5, −6, and 7?
A.) f(x) = x3 − 6x2 − 30x + 37
B.) f(x) = x3 − 30x2 − 37x + 42
C.) f(x) = x3 − 6x2 − 37x + 210
D.) f(x) = x3 − 30x2 − 42x + 210

Answer 1

(x - 5)(x + 6)(x - 7) = 0 
(x^2 + x - 30)(x - 7) = 0 
x^3 + x^2 - 30x - 7x^2 - 7x + 210 
x^3 - 6x^2 - 37x + 210 
ANSWER- C

Answer 2

Given roots of the polynomial :  5, −6, and 7.

Therefore, zeros are x=5, x=-6 and x=7.

So, for the given zeros, factors are

(x-5)(x+6)(x-7).

Now, we need to multiply (x-5)(x+6)(x-7) those factors to get the final polynomial.

$\mathrm{Expand}\:\left(x-5\right)\left(x+6\right)= x^2+6x-5x-30$

$=x^2+x-30$

$\left(x-5\right)\left(x+6\right)\left(x-7\right)=\left(x^2+x-30\right)\left(x-7\right)$

$\mathrm{Expand}\:\left(x^2+x-30\right)\left(x-7\right)$

$=x^3-7x^2+x^2-7x-30x+30$

$=x^3-6x^2-37x+210.$

Therefore, correct option is :

$C.) f(x) = x^3- 6x^2-37x + 210$