Given polynomial f(X) and a factor of f(x), factor f(x) completely.
F(x)= x^3-9x^2+8x+60; x+2
dividing the polynomial by the factor gives ... x^2 - 11x + 30
factor the quotient to find the other two factors of F(x)
To factor the polynomial f(x) = x^3 - 9x^2 + 8x + 60 completely, given the factor x + 2, we can use the synthetic division method. Here are the step-by-step instructions:
Step 1: Write down the coefficients of the polynomial f(x) in order. The coefficients for f(x) = x^3 - 9x^2 + 8x + 60 are 1, -9, 8, and 60.
Step 2: Set up the synthetic division table, with the factor x + 2 on the left side and the coefficients on the top.
-2 │ 1 -9 8 60
│______________
Step 3: Bring down the first coefficient (1) to the bottom row.
-2 │ 1 -9 8 60
│______________
1
Step 4: Multiply the factor (-2) by the value in the bottom row, and write the result in the next column.
-2 │ 1 -9 8 60
│______________
1 2
Step 5: Add the values in the second column (9 + 2 = 11), and write the result in the next column.
-2 │ 1 -9 8 60
│______________
1 2 11
Step 6: Multiply the factor (-2) by the value in the third column, and write the result in the next column.
-2 │ 1 -9 8 60
│______________
1 2 11 -38
Step 7: Add the values in the fourth column (8 + (-38) = -30), and write the result in the next column.
-2 │ 1 -9 8 60
│______________
1 2 11 -38
-30
Step 8: The coefficients in the bottom row represent the quotient of the polynomial division. The polynomial can be factored as follows:
f(x) = (x + 2)(x^2 + 2x - 30)
So, the factored form of f(x) = x^3 - 9x^2 + 8x + 60, with the factor x + 2, is (x + 2)(x^2 + 2x - 30).
To factorize the given polynomial, f(x) = x^3 - 9x^2 + 8x + 60, using the given factor x + 2, you can follow these steps:
Step 1: Use the factor theorem to check if x + 2 is a zero of f(x).
The factor theorem states that if f(a) = 0, then (x - a) is a factor of f(x).
Substitute -2 in place of x in f(x):
f(-2) = (-2)^3 - 9(-2)^2 + 8(-2) + 60
= -8 - 36 - 16 + 60
= 0
Since f(-2) = 0, this confirms that x + 2 is indeed a factor of f(x).
Step 2: Use synthetic division to divide f(x) by x + 2.
The coefficients of f(x) are 1, -9, 8, and 60.
Set up the synthetic division table:
| -2 | 1 | -9 | 8 | 60 |
| | | | | |
Bring down the first coefficient, which is 1, and multiply it by -2 to get -2:
| -2 | 1 | -9 | 8 | 60 |
| | | | | |
1
Add -2 and -9 to get -11, then multiply it by -2 to get 22:
| -2 | 1 | -9 | 8 | 60 |
| | | | | |
1 -2
Add 22 and 8 to get 30, then multiply it by -2 to get -60:
| -2 | 1 | -9 | 8 | 60 |
| | | | | |
1 -2 22
Add -60 and 60 to get 0.
The final row of coefficients, 1, -2, 22, 0, represents the quotient of the division.
Therefore, the factored form of f(x) is:
f(x) = (x + 2)(x^2 - 2x + 22)