Which of the following represents the zeros of the function g(x) = x 3 - 9x 2 + 2x + 48?
A. x = –8, x = 2, and x = –3
B. x = 8, x = –2, and x = 3
C. x = 6, x = –4, and x = 9
D. x = –6, x = 4, and x = –9
I need help
To find the zeros of the function g(x) = x^3 - 9x^2 + 2x + 48, we need to find the values of x for which g(x) equals zero.
One way to do this is by factoring the expression. Unfortunately, factoring cubic expressions can be challenging and may not always be possible. In this case, factoring may not be straightforward.
Another method is to use numerical methods, such as the Newton-Raphson method or graphing the function to find the x-intercepts. However, these methods can be time-consuming and may not give exact solutions.
A more efficient and accurate method is to use synthetic division or long division to divide the given polynomial by the possible values for x and check for remainders equal to zero. For example, to check if x = -8 is a zero of g(x), we perform synthetic division or long division using -8 as the divisor and check if the remainder is zero. We continue this process for all the given options until we find the values that give a remainder of zero.
Let's begin by checking option A: x = -8, x = 2, and x = -3.
Using synthetic division, we divide g(x) by (x + 8) and check if the remainder is zero:
-8 | 1 -9 2 48
-8 136 -544
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1 -17 138 -496
Since the remainder is not zero, x = -8 is not a zero of the function.
Next, we check x = 2:
2 | 1 -9 2 48
2 -14 -24
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1 -7 -12 24
Again, the remainder is not zero, so x = 2 is not a zero of the function.
Finally, we check x = -3:
-3 | 1 -9 2 48
-3 36 -114
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1 -12 38 -66
The remainder is not zero, so x = -3 is not a zero of the function.
Therefore, none of the options provided (A, B, C, or D) represents the zeros of the function g(x) = x^3 - 9x^2 + 2x + 48.
If none of the answer choices match the zeros, there may be a mistake in the given options or the function itself. Double-checking the problem statement and recalculating the function may help to verify the correct zeros.
g(x) = x^3 - 9x^2 + 2x + 48
Look for the easy roots first: 1,2,3.
A little synthetic division quickly shows that
g(x) = (x+2)(x-3)(x-8)