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Answer :
To solve the polynomial division problem [tex]\((-99 - 40x + 4x^5 + 18x^4 - 70x^3) \div (4x - 10)\)[/tex], we need to perform polynomial long division. Here’s a step-by-step explanation of the process:
1. Set Up the Division:
- Divide the polynomial [tex]\(4x^5 + 18x^4 - 70x^3 - 40x - 99\)[/tex] by the divisor [tex]\(4x - 10\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(4x^5\)[/tex], by the leading term of the divisor, [tex]\(4x\)[/tex]. This gives [tex]\(x^4\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(x^4\)[/tex] by the entire divisor [tex]\(4x - 10\)[/tex], which gives [tex]\(4x^5 - 10x^4\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(4x^5 + 18x^4 - 70x^3 - 40x - 99) - (4x^5 - 10x^4) = 28x^4 - 70x^3 - 40x - 99
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\(28x^4\)[/tex] by the leading term [tex]\(4x\)[/tex] to get [tex]\(7x^3\)[/tex].
- Multiply [tex]\(7x^3\)[/tex] by [tex]\(4x - 10\)[/tex] to get [tex]\(28x^4 - 70x^3\)[/tex].
- Subtract again:
[tex]\[
(28x^4 - 70x^3 - 40x - 99) - (28x^4 - 70x^3) = -40x - 99
\][/tex]
5. Continue Until the End:
- Now divide [tex]\(-40x\)[/tex] by [tex]\(4x\)[/tex] to obtain [tex]\(-10\)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\(4x - 10\)[/tex] to get [tex]\(-40x + 100\)[/tex].
- Subtract again:
[tex]\[
(-40x - 99) - (-40x + 100) = -199
\][/tex]
6. Assemble the Result:
- The quotient of the division is [tex]\(x^4 + 7x^3 - 10\)[/tex].
- The remainder is [tex]\(-199\)[/tex].
In summary, when dividing the polynomial [tex]\(4x^5 + 18x^4 - 70x^3 - 40x - 99\)[/tex] by [tex]\(4x - 10\)[/tex], we obtain a quotient of [tex]\(x^4 + 7x^3 - 10\)[/tex] and a remainder of [tex]\(-199\)[/tex].
1. Set Up the Division:
- Divide the polynomial [tex]\(4x^5 + 18x^4 - 70x^3 - 40x - 99\)[/tex] by the divisor [tex]\(4x - 10\)[/tex].
2. Divide the Leading Terms:
- Divide the leading term of the dividend, [tex]\(4x^5\)[/tex], by the leading term of the divisor, [tex]\(4x\)[/tex]. This gives [tex]\(x^4\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(x^4\)[/tex] by the entire divisor [tex]\(4x - 10\)[/tex], which gives [tex]\(4x^5 - 10x^4\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(4x^5 + 18x^4 - 70x^3 - 40x - 99) - (4x^5 - 10x^4) = 28x^4 - 70x^3 - 40x - 99
\][/tex]
4. Repeat the Process:
- Divide the new leading term [tex]\(28x^4\)[/tex] by the leading term [tex]\(4x\)[/tex] to get [tex]\(7x^3\)[/tex].
- Multiply [tex]\(7x^3\)[/tex] by [tex]\(4x - 10\)[/tex] to get [tex]\(28x^4 - 70x^3\)[/tex].
- Subtract again:
[tex]\[
(28x^4 - 70x^3 - 40x - 99) - (28x^4 - 70x^3) = -40x - 99
\][/tex]
5. Continue Until the End:
- Now divide [tex]\(-40x\)[/tex] by [tex]\(4x\)[/tex] to obtain [tex]\(-10\)[/tex].
- Multiply [tex]\(-10\)[/tex] by [tex]\(4x - 10\)[/tex] to get [tex]\(-40x + 100\)[/tex].
- Subtract again:
[tex]\[
(-40x - 99) - (-40x + 100) = -199
\][/tex]
6. Assemble the Result:
- The quotient of the division is [tex]\(x^4 + 7x^3 - 10\)[/tex].
- The remainder is [tex]\(-199\)[/tex].
In summary, when dividing the polynomial [tex]\(4x^5 + 18x^4 - 70x^3 - 40x - 99\)[/tex] by [tex]\(4x - 10\)[/tex], we obtain a quotient of [tex]\(x^4 + 7x^3 - 10\)[/tex] and a remainder of [tex]\(-199\)[/tex].
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