which of the following is true about the polynomial f(x) = x3 + 9x2 + 24x + 16 ?
a. -1 is a zero
b. 1 is a zero
c. -2 is a zero
d. 2i is a zero
f(-2)=-8+36-48+16=not zero
f(-1)=-1+9-24+16= zero, it is a zero.
To determine which of the given options is true about the polynomial f(x) = x^3 + 9x^2 + 24x + 16, we can use the concept of zeros or roots.
The zeros of a polynomial are the values of x for which the polynomial evaluates to zero. In other words, if x is a zero of the polynomial, then f(x) = 0.
To verify the given options, we need to check if substituting the value for x in the polynomial results in f(x) = 0.
a. Checking if -1 is a zero:
Substitute x = -1 into the polynomial:
f(-1) = (-1)^3 + 9(-1)^2 + 24(-1) + 16
= -1 + 9 - 24 + 16
= 0
Therefore, -1 is a zero of the polynomial f(x) = x^3 + 9x^2 + 24x + 16.
b. Checking if 1 is a zero:
Substitute x = 1 into the polynomial:
f(1) = (1)^3 + 9(1)^2 + 24(1) + 16
= 1 + 9 + 24 + 16
= 50
Therefore, 1 is not a zero of the polynomial f(x) = x^3 + 9x^2 + 24x + 16.
c. Checking if -2 is a zero:
Substitute x = -2 into the polynomial:
f(-2) = (-2)^3 + 9(-2)^2 + 24(-2) + 16
= -8 + 36 - 48 + 16
= -4
Therefore, -2 is not a zero of the polynomial f(x) = x^3 + 9x^2 + 24x + 16.
d. Checking if 2i is a zero (where i represents the imaginary unit):
Substitute x = 2i into the polynomial:
f(2i) = (2i)^3 + 9(2i)^2 + 24(2i) + 16
= 8i^3 + 36i^2 + 48i + 16
However, since i^2 = -1 and i^3 = -i, we can simplify the expression:
f(2i) = 8(-i) + 36(-1) + 48i + 16
= -8i - 36 + 48i + 16
= 40i - 20
= 20(2i - 1)
Since the expression 2i - 1 does not equal zero, 2i is not a zero of the polynomial f(x) = x^3 + 9x^2 + 24x + 16.
In conclusion, the correct answer is:
a. -1 is a zero