the polynomial y= x^3 + 9x^2 - 30x - 270 has an x-intercept at x=-9
a) use the x-intercept to find a factor of the polynomial
b) write the polynomial in factored form
c) identify the exact value of any other x-intercepts of the polynomial
a) Since x = -9 is an x-intercept, it means that (x + 9) is a factor of the polynomial.
b) To factor the polynomial, we divide it by (x + 9) using long division or synthetic division:
(x^3 + 9x^2 - 30x - 270) / (x + 9) = x^2 + 0x - 30
Therefore, the factored form of the polynomial is: (x + 9)(x^2 - 30)
c) To find the other x-intercepts, we set the factored form of the polynomial equal to zero and solve for x:
(x + 9)(x^2 - 30) = 0
Setting each factor to zero:
x + 9 = 0 -> x = -9 (x-intercept given)
x^2 - 30 = 0 -> x^2 = 30 -> x = ±√30
So the other x-intercepts of the polynomial are x = -9 and x = ±√30.