Answer :
To divide the polynomial [tex]\(x^4 - 6x^3 + 10x + 36\)[/tex] by [tex]\(x - 3\)[/tex], we can use polynomial long division. Here’s a detailed step-by-step solution:
1. Setup the Division: We're dividing [tex]\(x^4 - 6x^3 + 0x^2 + 10x + 36\)[/tex] by [tex]\(x - 3\)[/tex]. Note that we included [tex]\(0x^2\)[/tex] to account for the missing [tex]\(x^2\)[/tex] term for easier alignment during division.
2. Divide the First Term: Divide the leading term of the dividend (which is [tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]). This gives you [tex]\(x^3\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(x^3\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex], which gives [tex]\(x^4 - 3x^3\)[/tex]. Subtract this result from the original dividend:
[tex]\[
(x^4 - 6x^3 + 0x^2 + 10x + 36) - (x^4 - 3x^3) = -3x^3 + 0x^2 + 10x + 36
\][/tex]
4. Repeat the Process: Now, divide [tex]\(-3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3x^2\)[/tex].
5. Multiply and Subtract Again: Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(-3x^3 + 9x^2\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(-3x^3 + 0x^2 + 10x + 36) - (-3x^3 + 9x^2) = -9x^2 + 10x + 36
\][/tex]
6. Continue the Process: Divide [tex]\(-9x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-9x\)[/tex].
7. Multiply and Subtract Again: Multiply [tex]\(-9x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(-9x^2 + 27x\)[/tex]. Subtract:
[tex]\[
(-9x^2 + 10x + 36) - (-9x^2 + 27x) = -17x + 36
\][/tex]
8. Last Step: Divide [tex]\(-17x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-17\)[/tex].
9. Multiply and Subtract Once More: Multiply [tex]\(-17\)[/tex] by [tex]\(x - 3\)[/tex] giving [tex]\(-17x + 51\)[/tex]. Subtract:
[tex]\[
(-17x + 36) - (-17x + 51) = -15
\][/tex]
The division is complete as there are no more terms to bring down. The quotient is [tex]\(x^3 - 3x^2 - 9x - 17\)[/tex] and the remainder is [tex]\(-15\)[/tex].
Hence, the final result is:
[tex]\[
x^4 - 6x^3 + 10x + 36 \div (x - 3) = x^3 - 3x^2 - 9x - 17 \text{ with a remainder of } -15
\][/tex]
1. Setup the Division: We're dividing [tex]\(x^4 - 6x^3 + 0x^2 + 10x + 36\)[/tex] by [tex]\(x - 3\)[/tex]. Note that we included [tex]\(0x^2\)[/tex] to account for the missing [tex]\(x^2\)[/tex] term for easier alignment during division.
2. Divide the First Term: Divide the leading term of the dividend (which is [tex]\(x^4\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]). This gives you [tex]\(x^3\)[/tex].
3. Multiply and Subtract: Multiply [tex]\(x^3\)[/tex] by the entire divisor [tex]\(x - 3\)[/tex], which gives [tex]\(x^4 - 3x^3\)[/tex]. Subtract this result from the original dividend:
[tex]\[
(x^4 - 6x^3 + 0x^2 + 10x + 36) - (x^4 - 3x^3) = -3x^3 + 0x^2 + 10x + 36
\][/tex]
4. Repeat the Process: Now, divide [tex]\(-3x^3\)[/tex] by [tex]\(x\)[/tex], which gives [tex]\(-3x^2\)[/tex].
5. Multiply and Subtract Again: Multiply [tex]\(-3x^2\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(-3x^3 + 9x^2\)[/tex]. Subtract this from the current polynomial:
[tex]\[
(-3x^3 + 0x^2 + 10x + 36) - (-3x^3 + 9x^2) = -9x^2 + 10x + 36
\][/tex]
6. Continue the Process: Divide [tex]\(-9x^2\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-9x\)[/tex].
7. Multiply and Subtract Again: Multiply [tex]\(-9x\)[/tex] by [tex]\(x - 3\)[/tex] to get [tex]\(-9x^2 + 27x\)[/tex]. Subtract:
[tex]\[
(-9x^2 + 10x + 36) - (-9x^2 + 27x) = -17x + 36
\][/tex]
8. Last Step: Divide [tex]\(-17x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-17\)[/tex].
9. Multiply and Subtract Once More: Multiply [tex]\(-17\)[/tex] by [tex]\(x - 3\)[/tex] giving [tex]\(-17x + 51\)[/tex]. Subtract:
[tex]\[
(-17x + 36) - (-17x + 51) = -15
\][/tex]
The division is complete as there are no more terms to bring down. The quotient is [tex]\(x^3 - 3x^2 - 9x - 17\)[/tex] and the remainder is [tex]\(-15\)[/tex].
Hence, the final result is:
[tex]\[
x^4 - 6x^3 + 10x + 36 \div (x - 3) = x^3 - 3x^2 - 9x - 17 \text{ with a remainder of } -15
\][/tex]
