To divide the polynomial [tex]\( \left(x^3 - 3x^2 + 2x - 1\right) \)[/tex] by [tex]\( \left(x^2 - 7\right) \)[/tex], we'll perform polynomial long division.
Here are the step-by-step instructions:
1. Setup the Division:
```
Divide: (x^3 - 3x^2 + 2x - 1) by (x^2 - 7)
```
2. Division Step:
- Divide the leading term of the numerator [tex]\(x^3\)[/tex] by the leading term of the denominator [tex]\(x^2\)[/tex] to get the first term of the quotient.
- [tex]\( x^3 \div x^2 = x \)[/tex]
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\(x^2 - 7\)[/tex] by [tex]\(x\)[/tex] (the first term of the quotient):
[tex]\[
x(x^2 - 7) = x^3 - 7x
\][/tex]
- Subtract this product from the current dividend:
[tex]\[
(x^3 - 3x^2 + 2x - 1) - (x^3 - 7x) = -3x^2 + 2x - 1 -(-7x) = -3x^2 + 9x - 1
\][/tex]
4. Next Term of the Quotient:
- Divide the new leading term of the current dividend [tex]\(-3x^2\)[/tex] by the leading term of the divisor [tex]\(x^2\)[/tex] to get the next term of the quotient:
[tex]\[
-3x^2 \div x^2 = -3
\][/tex]
5. Multiply and Subtract Again:
- Multiply the entire divisor [tex]\(x^2 - 7\)[/tex] by [tex]\(-3\)[/tex] (the next term of the quotient):
[tex]\[
-3(x^2 - 7) = -3x^2 + 21
\][/tex]
- Subtract this product from the current dividend:
[tex]\[
(-3x^2 + 9x - 1) - (-3x^2 + 21) = 9x - 1 - 21 = 9x - 22
\][/tex]
6. Result:
- The quotient from the division is [tex]\(x - 3\)[/tex].
- The remainder of the division is [tex]\(9x - 22\)[/tex].
So, the division of [tex]\(\left(x^3 - 3x^2 + 2x - 1\right) \div \left(x^2 - 7\right)\)[/tex] yields:
[tex]\[
\text{Quotient} = x - 3, \quad \text{Remainder} = 9x - 22
\][/tex]
Final answer:
[tex]\[
x^3 - 3x^2 + 2x - 1 = (x^2 - 7)(x - 3) + 9x - 22
\][/tex]
Thus,
[tex]\[
\left( x^3 - 3 x^2 + 2 x - 1 \right) \div \left( x^2 - 7 \right) = x - 3 + \frac{9x - 22}{x^2 - 7}
\][/tex]
