Answer :
Sure! Let's solve the problem of dividing the polynomial [tex]\(2x^6 + 6x^4 - 7\)[/tex] by [tex]\(x^3 + x^2 - 7\)[/tex] using polynomial long division. Here's a step-by-step process:
### Step 1: Set up the division
- Dividend: [tex]\(2x^6 + 0x^5 + 6x^4 + 0x^3 + 0x^2 + 0x - 7\)[/tex]
- Divisor: [tex]\(x^3 + x^2 - 7\)[/tex]
### Step 2: Perform the division
1st Term:
- Divide the leading term of the dividend [tex]\(2x^6\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^6}{x^3} = 2x^3
\][/tex]
- Multiply the entire divisor by [tex]\(2x^3\)[/tex] and subtract from the dividend:
[tex]\[
(2x^6 + 2x^5 - 14x^3)
\][/tex]
- Subtract from the dividend:
[tex]\[
(2x^6 + 0x^5 + 6x^4 + 0x^3 + 0x^2 + 0x - 7) - (2x^6 + 2x^5 - 14x^3) = -2x^5 + 6x^4 + 14x^3 + 0x^2 + 0x - 7
\][/tex]
2nd Term:
- Divide [tex]\(-2x^5\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-2x^5}{x^3} = -2x^2
\][/tex]
- Multiply the entire divisor by [tex]\(-2x^2\)[/tex] and subtract:
[tex]\[
(-2x^5 - 2x^4 + 14x^2)
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(-2x^5 + 6x^4 + 14x^3 + 0x^2 + 0x - 7) - (-2x^5 - 2x^4 + 14x^2) = 8x^4 + 14x^3 - 14x^2 + 0x - 7
\][/tex]
3rd Term:
- Divide [tex]\(8x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{8x^4}{x^3} = 8x
\][/tex]
- Multiply the entire divisor by [tex]\(8x\)[/tex] and subtract:
[tex]\[
(8x^4 + 8x^3 - 56x)
\][/tex]
- Subtract:
[tex]\[
(8x^4 + 14x^3 - 14x^2 + 0x - 7) - (8x^4 + 8x^3 - 56x) = 6x^3 - 14x^2 + 56x - 7
\][/tex]
4th Term:
- Divide [tex]\(6x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{6x^3}{x^3} = 6
\][/tex]
- Multiply the entire divisor by [tex]\(6\)[/tex] and subtract:
[tex]\[
(6x^3 + 6x^2 - 42)
\][/tex]
- Subtract:
[tex]\[
(6x^3 - 14x^2 + 56x - 7) - (6x^3 + 6x^2 - 42) = -20x^2 + 56x + 35
\][/tex]
### Conclusion
The quotient of the division is [tex]\(2x^3 - 2x^2 + 8x + 6\)[/tex], and the remainder is [tex]\(-20x^2 + 56x + 35\)[/tex].
So, the result of the division is:
[tex]\[
2x^3 - 2x^2 + 8x + 6 \quad \text{with remainder} \quad -20x^2 + 56x + 35
\][/tex]
### Step 1: Set up the division
- Dividend: [tex]\(2x^6 + 0x^5 + 6x^4 + 0x^3 + 0x^2 + 0x - 7\)[/tex]
- Divisor: [tex]\(x^3 + x^2 - 7\)[/tex]
### Step 2: Perform the division
1st Term:
- Divide the leading term of the dividend [tex]\(2x^6\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{2x^6}{x^3} = 2x^3
\][/tex]
- Multiply the entire divisor by [tex]\(2x^3\)[/tex] and subtract from the dividend:
[tex]\[
(2x^6 + 2x^5 - 14x^3)
\][/tex]
- Subtract from the dividend:
[tex]\[
(2x^6 + 0x^5 + 6x^4 + 0x^3 + 0x^2 + 0x - 7) - (2x^6 + 2x^5 - 14x^3) = -2x^5 + 6x^4 + 14x^3 + 0x^2 + 0x - 7
\][/tex]
2nd Term:
- Divide [tex]\(-2x^5\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{-2x^5}{x^3} = -2x^2
\][/tex]
- Multiply the entire divisor by [tex]\(-2x^2\)[/tex] and subtract:
[tex]\[
(-2x^5 - 2x^4 + 14x^2)
\][/tex]
- Subtract this result from the current dividend:
[tex]\[
(-2x^5 + 6x^4 + 14x^3 + 0x^2 + 0x - 7) - (-2x^5 - 2x^4 + 14x^2) = 8x^4 + 14x^3 - 14x^2 + 0x - 7
\][/tex]
3rd Term:
- Divide [tex]\(8x^4\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{8x^4}{x^3} = 8x
\][/tex]
- Multiply the entire divisor by [tex]\(8x\)[/tex] and subtract:
[tex]\[
(8x^4 + 8x^3 - 56x)
\][/tex]
- Subtract:
[tex]\[
(8x^4 + 14x^3 - 14x^2 + 0x - 7) - (8x^4 + 8x^3 - 56x) = 6x^3 - 14x^2 + 56x - 7
\][/tex]
4th Term:
- Divide [tex]\(6x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{6x^3}{x^3} = 6
\][/tex]
- Multiply the entire divisor by [tex]\(6\)[/tex] and subtract:
[tex]\[
(6x^3 + 6x^2 - 42)
\][/tex]
- Subtract:
[tex]\[
(6x^3 - 14x^2 + 56x - 7) - (6x^3 + 6x^2 - 42) = -20x^2 + 56x + 35
\][/tex]
### Conclusion
The quotient of the division is [tex]\(2x^3 - 2x^2 + 8x + 6\)[/tex], and the remainder is [tex]\(-20x^2 + 56x + 35\)[/tex].
So, the result of the division is:
[tex]\[
2x^3 - 2x^2 + 8x + 6 \quad \text{with remainder} \quad -20x^2 + 56x + 35
\][/tex]
