Answer :
To solve the problem of dividing the polynomial [tex]\(7x^3 + 52x^2 - 39x - 62\)[/tex] by [tex]\(x + 8\)[/tex], we can use polynomial long division. Here is how you can perform the division step by step:
1. Set up the division: Write [tex]\(7x^3 + 52x^2 - 39x - 62\)[/tex] as the dividend inside the division symbol and [tex]\(x + 8\)[/tex] as the divisor outside.
2. Divide the first term: Divide the leading term of the dividend [tex]\(7x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. This gives you [tex]\(7x^2\)[/tex].
3. Multiply and subtract: Multiply [tex]\(7x^2\)[/tex] by the entire divisor [tex]\(x + 8\)[/tex] to get [tex]\(7x^3 + 56x^2\)[/tex]. Subtract this from the dividend:
[tex]\[
(7x^3 + 52x^2 - 39x - 62) - (7x^3 + 56x^2) = -4x^2 - 39x - 62
\][/tex]
4. Repeat the process: Now take [tex]\(-4x^2\)[/tex] (the new leading term) and divide it by [tex]\(x\)[/tex]. This gives [tex]\(-4x\)[/tex].
5. Multiply and subtract again: Multiply [tex]\(-4x\)[/tex] by [tex]\(x + 8\)[/tex] to get [tex]\(-4x^2 - 32x\)[/tex]. Subtract this from the current dividend:
[tex]\[
(-4x^2 - 39x - 62) - (-4x^2 - 32x) = -7x - 62
\][/tex]
6. Repeat once more: Divide [tex]\(-7x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-7\)[/tex].
7. Final multiplication and subtraction: Multiply [tex]\(-7\)[/tex] by [tex]\(x + 8\)[/tex] to get [tex]\(-7x - 56\)[/tex]. Subtract this from what's left:
[tex]\[
(-7x - 62) - (-7x - 56) = -6
\][/tex]
After the division process is complete, the quotient is [tex]\(7x^2 - 4x - 7\)[/tex] and the remainder is [tex]\(-6\)[/tex].
So, when the polynomial [tex]\(7x^3 + 52x^2 - 39x - 62\)[/tex] is divided by [tex]\(x + 8\)[/tex], the result is:
[tex]\[
7x^2 - 4x - 7 \quad \text{with a remainder of} \quad -6
\][/tex]
1. Set up the division: Write [tex]\(7x^3 + 52x^2 - 39x - 62\)[/tex] as the dividend inside the division symbol and [tex]\(x + 8\)[/tex] as the divisor outside.
2. Divide the first term: Divide the leading term of the dividend [tex]\(7x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]. This gives you [tex]\(7x^2\)[/tex].
3. Multiply and subtract: Multiply [tex]\(7x^2\)[/tex] by the entire divisor [tex]\(x + 8\)[/tex] to get [tex]\(7x^3 + 56x^2\)[/tex]. Subtract this from the dividend:
[tex]\[
(7x^3 + 52x^2 - 39x - 62) - (7x^3 + 56x^2) = -4x^2 - 39x - 62
\][/tex]
4. Repeat the process: Now take [tex]\(-4x^2\)[/tex] (the new leading term) and divide it by [tex]\(x\)[/tex]. This gives [tex]\(-4x\)[/tex].
5. Multiply and subtract again: Multiply [tex]\(-4x\)[/tex] by [tex]\(x + 8\)[/tex] to get [tex]\(-4x^2 - 32x\)[/tex]. Subtract this from the current dividend:
[tex]\[
(-4x^2 - 39x - 62) - (-4x^2 - 32x) = -7x - 62
\][/tex]
6. Repeat once more: Divide [tex]\(-7x\)[/tex] by [tex]\(x\)[/tex] to get [tex]\(-7\)[/tex].
7. Final multiplication and subtraction: Multiply [tex]\(-7\)[/tex] by [tex]\(x + 8\)[/tex] to get [tex]\(-7x - 56\)[/tex]. Subtract this from what's left:
[tex]\[
(-7x - 62) - (-7x - 56) = -6
\][/tex]
After the division process is complete, the quotient is [tex]\(7x^2 - 4x - 7\)[/tex] and the remainder is [tex]\(-6\)[/tex].
So, when the polynomial [tex]\(7x^3 + 52x^2 - 39x - 62\)[/tex] is divided by [tex]\(x + 8\)[/tex], the result is:
[tex]\[
7x^2 - 4x - 7 \quad \text{with a remainder of} \quad -6
\][/tex]
