To perform synthetic division for the polynomial \( x^5 - 4x^3 + 2x + 6 \) by \( x + 2 \), we first rewrite \( x + 2 \) to find the root, which is \( -2 \).
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Set up the synthetic division:
- The coefficients of the polynomial \( x^5 - 4x^3 + 2x + 6 \) are:
- \( 1 \) (for \( x^5 \))
- \( 0 \) (for \( x^4 \), since there's no \( x^4 \) term)
- \( -4 \) (for \( x^3 \))
- \( 0 \) (for \( x^2 \), since there's no \( x^2 \) term)
- \( 2 \) (for \( x^1 \))
- \( 6 \) (for the constant term)
The coefficients are \([1, 0, -4, 0, 2, 6]\).
- The coefficients of the polynomial \( x^5 - 4x^3 + 2x + 6 \) are:
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Write down the synthetic division setup:
- The root \( -2 \) is placed on the left, and the coefficients are lined up on the right:
-2 | 1 0 -4 0 2 6 | |_________________________ -
Start the synthetic division process:
- Bring down the leading coefficient \( 1 \):
-2 | 1 0 -4 0 2 6 | |_________________________ 1- Now multiply \( 1 \) by \( -2 \) and add to the next coefficient \( 0 \):
-2 | 1 0 -4 0 2 6 | -2 |_________________________ 1 -2- Next, multiply \( -2 \) by \( -2 \) and add to the next coefficient \( -4 \):
-2 | 1 0 -4 0 2 6 | -2 4 |_________________________ 1 -2 0- Next, multiply \( 0 \) by \( -2 \) and add to the next coefficient \( 0 \):
-2 | 1 0 -4 0 2 6 | -2 4 0 |_________________________ 1 -2 0 2- Next, multiply \( 2 \) by \( -2 \) and add to the next coefficient \( 6 \):
-2 | 1 0 -4 0 2 6 | -2 4 0 -4 |_________________________ 1 -2 0 2 2 -
Finalize the synthetic division:
- The resulting row represents the coefficients of the quotient polynomial and the remainder. The quotient polynomial is: \[ x^4 - 2x^3 + 0x^2 + 2x + 2 \]
- The remainder is \( 2 \).
Thus, the final result of the synthetic division is: \[ \text{Quotient: } x^4 - 2x^3 + 2x + 2 \quad \text{Remainder: } 2 \]
To summarize: \[ \frac{x^5 - 4x^3 + 2x + 6}{x + 2} = x^4 - 2x^3 + 2x + 2 + \frac{2}{x + 2} \]
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