Factorise as a product of a linear and quadratic factor. (x+2) is a factor
(X^3-2x^2+16)
To factorize the expression (x^3 - 2x^2 + 16) by using the given factor (x + 2), we can follow these steps:
Step 1: Apply synthetic division to divide the given expression by (x + 2).
- Write the factor (x + 2) as (-2) since (x + 2) = 0 gives x = -2.
- Set up the synthetic division using the coefficients of the terms in the expression:
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-2 | 1 -2 0 16
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Step 2: Perform the synthetic division to get the quotient.
- Bring down the first coefficient (1) and multiply by the divisor (-2) to get -2.
- Add -2 to the second coefficient (-2) to get -4.
- Multiply -4 by -2 to get 8, and add it to the third coefficient (0) to get 8.
- Multiply 8 by -2 to get -16, and add it to the fourth coefficient (16) to get 0.
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-2 | 1 -2 0 16
-|___________-2 4 -8
1 -4 4 8
- The quotient is 1x² - 4x + 4, and the remainder is 8.
Step 3: Write the expression in factored form using the quotient obtained from the synthetic division.
- The quotient is 1x² - 4x + 4.
- Rearrange the terms to obtain: x² - 4x + 4.
Step 4: Factorize the quadratic expression.
- The quadratic expression x² - 4x + 4 can be factored as a perfect square: (x - 2)(x - 2).
- Therefore, the factored form of the expression (x^3 - 2x^2 + 16) using (x + 2) is (x + 2)(x - 2)(x - 2).
So, (x^3 - 2x^2 + 16) can be expressed as a product of a linear factor (x + 2) and quadratic factors (x - 2) and (x - 2).