To perform synthetic division for the polynomial division \( 8x^3 + 6x^2 - 18x + 4 \) by \( 4x - 1 \), we first need to convert the divisor \( 4x - 1 \) into the form \( x - r \). We find \( r \) by setting \( 4x - 1 = 0 \):
\[ 4x - 1 = 0 \implies 4x = 1 \implies x = \frac{1}{4} \]
Now we can use synthetic division with \( r = \frac{1}{4} \). Here are the steps to perform synthetic division:
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Write down the coefficients of the dividend. The polynomial \( 8x^3 + 6x^2 - 18x + 4 \) has the coefficients \( 8, 6, -18, 4 \).
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Set up synthetic division as follows, placing \( \frac{1}{4} \) on the left and the coefficients on the right:
1/4 | 8 6 -18 4
|
- Bring down the leading coefficient:
1/4 | 8 6 -18 4
|
|___________________
8
- Multiply \( 8 \) by \( \frac{1}{4} \) and write the result under the next coefficient:
1/4 | 8 6 -18 4
| 2
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8 8
- Add the second column:
1/4 | 8 6 -18 4
| 2
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8 8
- Continue this process. Multiply \( 8 \) by \( \frac{1}{4} \) and add to the next coefficient:
1/4 | 8 6 -18 4
| 2 2
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8 8 -16
- Multiply \( -16 \) by \( \frac{1}{4} \) and add to \( 4 \):
1/4 | 8 6 -18 4
| 2 2 -4
|___________________
8 8 -16 0
- The result shows the coefficients of the quotient polynomial and the remainder. The quotient is \( 8x^2 + 8x - 16 \) and the remainder is \( 0 \).
So, the result of the division \( 8x^3 + 6x^2 - 18x + 4 \) divided by \( 4x - 1 \) is:
\[ 8x^2 + 8x - 16 \]
with no remainder.
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