use factor theorem to determine whether x+2 is a factor of f(x). if it is, write f(x) in factored form.
f(x)=9x^3+24x^2+19x+14; x+2
since f(-2) = 0, (x+2) is a factor
f(x) = (x+2)(9x^2+6x+7)
To determine whether x + 2 is a factor of f(x) using the factor theorem, we need to perform synthetic division. Here's how you can do it:
Step 1: Set up the synthetic division:
-2 | 9 24 19 14
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Step 2: Bring down the first coefficient of f(x), which is 9:
-2 | 9 24 19 14
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9
Step 3: Multiply the result (9) by -2 and add it to the next coefficient:
-2 | 9 24 19 14
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9
-18
Step 4: Repeat step 3 until all coefficients are evaluated:
-2 | 9 24 19 14
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9
-18 -12
-6
Step 5: Read the last value in the synthetic division row. If it is equal to zero, then x + 2 is a factor of f(x). Otherwise, it is not.
In this case, the last value is -6, which is not zero. Therefore, x + 2 is not a factor of f(x).
To write f(x) in factored form, we can use the quadratic equation to solve for the remaining factors. However, since x + 2 is not a factor, f(x) cannot be fully factored using simple linear factors.