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Answer :
To divide the polynomial [tex]\(-35c^3 + 13c^2 + 46c - 26\)[/tex] by [tex]\(7c - 4\)[/tex] using synthetic division, follow these steps:
1. Identify the root of the divisor:
The divisor is [tex]\(7c - 4\)[/tex]. To use synthetic division, we first find the root by setting the divisor equal to zero:
[tex]\[
7c - 4 = 0 \implies c = \frac{4}{7}
\][/tex]
2. Set up the synthetic division:
- Write the coefficients of the polynomial to be divided: [tex]\(-35, 13, 46, -26\)[/tex].
- The root [tex]\(c = \frac{4}{7}\)[/tex] will be used to perform synthetic division.
3. Perform synthetic division:
- Start with the first coefficient: [tex]\(-35\)[/tex]. Since [tex]\(35\)[/tex] is the initial value, bring it down.
- Multiply [tex]\(-35\)[/tex] by the root [tex]\(\frac{4}{7}\)[/tex] and add the result to the next coefficient:
[tex]\[
-35 \times \frac{4}{7} = -\frac{140}{7} = -20
\][/tex]
Add [tex]\(-20\)[/tex] to [tex]\(13\)[/tex]:
[tex]\[
13 - 20 = -7
\][/tex]
- Now, use [tex]\(-7\)[/tex] in the second row:
[tex]\[
-7 \times \frac{4}{7} = -4
\][/tex]
Add [tex]\(-4\)[/tex] to [tex]\(46\)[/tex]:
[tex]\[
46 - 4 = 42
\][/tex]
- Finally, use [tex]\(42\)[/tex]:
[tex]\[
42 \times \frac{4}{7} = 24
\][/tex]
Add [tex]\(24\)[/tex] to [tex]\(-26\)[/tex]:
[tex]\[
-26 + 24 = -2
\][/tex]
4. Write down the results:
- The coefficients of the quotient polynomial are [tex]\([-35, -7, 42]\)[/tex], corresponding to [tex]\(-35c^2 - 7c + 42\)[/tex].
- The remainder is [tex]\(-2\)[/tex].
Therefore, the result of the synthetic division is:
[tex]\[
\frac{-35c^3 + 13c^2 + 46c - 26}{7c - 4} = -35c^2 - 7c + 42 \quad \text{with a remainder of } -2
\][/tex]
This means:
[tex]\[
-35c^3 + 13c^2 + 46c - 26 = (7c - 4)(-35c^2 - 7c + 42) - 2
\][/tex]
1. Identify the root of the divisor:
The divisor is [tex]\(7c - 4\)[/tex]. To use synthetic division, we first find the root by setting the divisor equal to zero:
[tex]\[
7c - 4 = 0 \implies c = \frac{4}{7}
\][/tex]
2. Set up the synthetic division:
- Write the coefficients of the polynomial to be divided: [tex]\(-35, 13, 46, -26\)[/tex].
- The root [tex]\(c = \frac{4}{7}\)[/tex] will be used to perform synthetic division.
3. Perform synthetic division:
- Start with the first coefficient: [tex]\(-35\)[/tex]. Since [tex]\(35\)[/tex] is the initial value, bring it down.
- Multiply [tex]\(-35\)[/tex] by the root [tex]\(\frac{4}{7}\)[/tex] and add the result to the next coefficient:
[tex]\[
-35 \times \frac{4}{7} = -\frac{140}{7} = -20
\][/tex]
Add [tex]\(-20\)[/tex] to [tex]\(13\)[/tex]:
[tex]\[
13 - 20 = -7
\][/tex]
- Now, use [tex]\(-7\)[/tex] in the second row:
[tex]\[
-7 \times \frac{4}{7} = -4
\][/tex]
Add [tex]\(-4\)[/tex] to [tex]\(46\)[/tex]:
[tex]\[
46 - 4 = 42
\][/tex]
- Finally, use [tex]\(42\)[/tex]:
[tex]\[
42 \times \frac{4}{7} = 24
\][/tex]
Add [tex]\(24\)[/tex] to [tex]\(-26\)[/tex]:
[tex]\[
-26 + 24 = -2
\][/tex]
4. Write down the results:
- The coefficients of the quotient polynomial are [tex]\([-35, -7, 42]\)[/tex], corresponding to [tex]\(-35c^2 - 7c + 42\)[/tex].
- The remainder is [tex]\(-2\)[/tex].
Therefore, the result of the synthetic division is:
[tex]\[
\frac{-35c^3 + 13c^2 + 46c - 26}{7c - 4} = -35c^2 - 7c + 42 \quad \text{with a remainder of } -2
\][/tex]
This means:
[tex]\[
-35c^3 + 13c^2 + 46c - 26 = (7c - 4)(-35c^2 - 7c + 42) - 2
\][/tex]
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