Find all the zeros of the following function.
g(x) = x^3 + 10x^2 + 28x + 49
a. -7
b. ± 1, ± 7, ± 49
c. 7, -3+i√19/2, -3-i√19/2
d. -7, -3+i√19/2, -3-i√19/2
To find the zeros of the function g(x), we need to find the values of x for which g(x) = 0.
To do this, we can either factor the polynomial or use synthetic division.
Unfortunately, the polynomial x^3 + 10x^2 + 28x + 49 cannot be factored easily. Therefore, we will use synthetic division to test for roots.
Let's start by testing x = -7 as a possible root using synthetic division:
-7 | 1 10 28 49
-7 -21 -49
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1 3 7 0
Since the remainder is 0, we know that x = -7 is a zero of the function.
Now, we can use synthetic division again to divide the resulting polynomial (1x^2 + 3x + 7) by (x + 7):
-7 | 1 3 7
-7 28
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1 -4 0
This gives us the quadratic equation 1x^2 - 4x = 0. Factoring out x, we get x(x - 4) = 0. This means that x = 0 and x = 4 are both zeros of the function.
Therefore, the correct answer is:
d. -7, -3+i√19/2, -3-i√19/2