Answer :
To solve the division of the polynomial [tex]\(x^4 - 70x^3 + 79x^2 + 9x - 7\)[/tex] by [tex]\(x^2 - 4x + 7\)[/tex], we will perform polynomial long division. Here's how you can do it step by step:
1. Start with the leading terms:
- Divide the leading term of the numerator [tex]\(x^4\)[/tex] by the leading term of the denominator [tex]\(x^2\)[/tex], which gives [tex]\(x^2\)[/tex].
2. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^2 - 4x + 7\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(x^4 - 4x^3 + 7x^2\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(x^4 - 70x^3 + 79x^2 + 9x - 7) - (x^4 - 4x^3 + 7x^2) = -66x^3 + 72x^2 + 9x - 7
\][/tex]
3. Repeat the process:
- Divide the new leading term [tex]\(-66x^3\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-66x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 - 4x + 7\)[/tex] by [tex]\(-66x\)[/tex] to get [tex]\(-66x^3 + 264x^2 - 462x\)[/tex].
- Subtract this result from the current polynomial:
[tex]\[
(-66x^3 + 72x^2 + 9x - 7) - (-66x^3 + 264x^2 - 462x) = -192x^2 + 471x - 7
\][/tex]
4. Continue the division:
- Divide the leading term [tex]\(-192x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-192\)[/tex].
- Multiply the entire divisor [tex]\(x^2 - 4x + 7\)[/tex] by [tex]\(-192\)[/tex] to get [tex]\(-192x^2 + 768x - 1344\)[/tex].
- Subtract this result from the current polynomial:
[tex]\[
(-192x^2 + 471x - 7) - (-192x^2 + 768x - 1344) = -297x + 1337
\][/tex]
5. Conclusion:
- The quotient is [tex]\(x^2 - 66x - 192\)[/tex].
- The remainder is [tex]\(-297x + 1337\)[/tex].
Thus, when [tex]\(x^4 - 70x^3 + 79x^2 + 9x - 7\)[/tex] is divided by [tex]\(x^2 - 4x + 7\)[/tex], the quotient is [tex]\(x^2 - 66x - 192\)[/tex], and the remainder is [tex]\(-297x + 1337\)[/tex].
1. Start with the leading terms:
- Divide the leading term of the numerator [tex]\(x^4\)[/tex] by the leading term of the denominator [tex]\(x^2\)[/tex], which gives [tex]\(x^2\)[/tex].
2. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^2 - 4x + 7\)[/tex] by [tex]\(x^2\)[/tex] to get [tex]\(x^4 - 4x^3 + 7x^2\)[/tex].
- Subtract this result from the original polynomial:
[tex]\[
(x^4 - 70x^3 + 79x^2 + 9x - 7) - (x^4 - 4x^3 + 7x^2) = -66x^3 + 72x^2 + 9x - 7
\][/tex]
3. Repeat the process:
- Divide the new leading term [tex]\(-66x^3\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-66x\)[/tex].
- Multiply the entire divisor [tex]\(x^2 - 4x + 7\)[/tex] by [tex]\(-66x\)[/tex] to get [tex]\(-66x^3 + 264x^2 - 462x\)[/tex].
- Subtract this result from the current polynomial:
[tex]\[
(-66x^3 + 72x^2 + 9x - 7) - (-66x^3 + 264x^2 - 462x) = -192x^2 + 471x - 7
\][/tex]
4. Continue the division:
- Divide the leading term [tex]\(-192x^2\)[/tex] by [tex]\(x^2\)[/tex], which gives [tex]\(-192\)[/tex].
- Multiply the entire divisor [tex]\(x^2 - 4x + 7\)[/tex] by [tex]\(-192\)[/tex] to get [tex]\(-192x^2 + 768x - 1344\)[/tex].
- Subtract this result from the current polynomial:
[tex]\[
(-192x^2 + 471x - 7) - (-192x^2 + 768x - 1344) = -297x + 1337
\][/tex]
5. Conclusion:
- The quotient is [tex]\(x^2 - 66x - 192\)[/tex].
- The remainder is [tex]\(-297x + 1337\)[/tex].
Thus, when [tex]\(x^4 - 70x^3 + 79x^2 + 9x - 7\)[/tex] is divided by [tex]\(x^2 - 4x + 7\)[/tex], the quotient is [tex]\(x^2 - 66x - 192\)[/tex], and the remainder is [tex]\(-297x + 1337\)[/tex].
