One of the zeros of the function f(x)=x^3-28x-48 is x=-4. Factor the function. Show your work.
How do I solve this?? Please help??
If x=-4 is one of the solutions, then x+4 must be a factor of x^3 - 28x - 48
so do a long division. You must get a quadratic with no remainder.
solve that quadratic = 0 to get the other two zeros.
hint: the remaining quadratic factors to give 2 more integer solutions.
To factor the function f(x) = x^3 - 28x - 48, we will use the fact that x = -4 is a zero of the function.
Step 1: Use synthetic division to divide f(x) by (x + 4).
-4 | 1 0 -28 -48
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Step-by-Step Solution:
To divide f(x) by (x + 4), we use synthetic division as follows:
First, arrange the coefficients of the function in descending order:
1 0 -28 -48
Start the synthetic division by bringing down the first coefficient, which is 1, to the bottom row:
-4 | 1 0 -28 -48
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1
Multiply the number at the bottom row (1) by the divisor (-4), then write the result in the second column:
-4 | 1 0 -28 -48
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1
Add the numbers in the second column:
-4 | 1 0 -28 -48
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1
Bring down the next coefficient, which is 0, and add it to the number in the third column:
-4 | 1 0 -28 -48
------
1
Multiply the number at the bottom row (1) by the divisor (-4), then write the result in the third column:
-4 | 1 0 -28 -48
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1 4
Add the numbers in the third column:
-4 | 1 0 -28 -48
------
1 4
Bring down the next coefficient, which is -28, and add it to the number in the fourth column:
-4 | 1 0 -28 -48
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1 4
Multiply the number at the bottom row (1) by the divisor (-4), then write the result in the fourth column:
-4 | 1 0 -28 -48
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1 4 112
Add the numbers in the fourth column:
-4 | 1 0 -28 -48
------
1 4 112
The numbers in the last row represent the coefficients of the quotient. In this case, the quotient is 1x^2 + 4x + 112.
Step 2: Write the factored form of the function.
From the synthetic division, we have:
f(x) = (x + 4)(x^2 + 4x + 112)
The factored form of the function f(x) = x^3 - 28x - 48 is (x + 4)(x^2 + 4x + 112).
To factor the given function f(x) = x^3 - 28x - 48, you can use synthetic division to divide the polynomial by the given zero x = -4. Here's how you can do it step by step:
Step 1: Set up synthetic division:
-4 | 1 0 -28 -48
Step 2: Bring down the coefficient of the highest degree term (the 1 in this case):
-4 | 1 0 -28 -48
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Step 3: Multiply the divisor (-4) by the first term (1), then write the result under the second term (0):
-4 | 1 0 -28 -48
-4
Step 4: Add the result obtained in step 3 to the second term (-4 + 0 = -4), write it below, then repeat the process:
-4 | 1 0 -28 -48
-4
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1
Step 5: Multiply the divisor (-4) by the result obtained in step 4 (1), then write the result below the third term (-28):
-4 | 1 0 -28 -48
-4 16
Step 6: Add the result obtained in step 5 to the third term (-28 + 16 = -12), write it below, then repeat the process:
-4 | 1 0 -28 -48
-4 16
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1 -12
Step 7: Multiply the divisor (-4) by the result obtained in step 6 (-12), then write the result below the fourth term (-48):
-4 | 1 0 -28 -48
-4 16 48
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Step 8: Add the result obtained in step 7 to the fourth term (-48 + 48 = 0), write it below:
-4 | 1 0 -28 -48
-4 16 48
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1 -12 20 0
Step 9: Read the coefficients of the quotient from the bottom row (1, -12, 20), which represents the factored form of the polynomial.
Therefore, the factored form of the function f(x) = x^3 - 28x - 48 is:
f(x) = (x + 4)(x^2 - 12x + 20)
Note: You can also verify your result by multiplying back the factors to see if you recover the original polynomial.