To solve the problem of dividing the polynomial [tex]\(x^3 - 2x^2 - 10x + 21\)[/tex] by [tex]\(x^2 + x - 7\)[/tex], we can use polynomial long division to find the quotient and remainder.
Here’s the step-by-step process:
1. Setup the Division:
[tex]\[
\begin{array}{r|l}
x^2 + x - 7 & x^3 - 2x^2 - 10x + 21 \\
\end{array}
\][/tex]
2. First Division Step:
- Divide the leading term of the numerator ([tex]\(x^3\)[/tex]) by the leading term of the denominator ([tex]\(x^2\)[/tex]):
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
- The first term of the quotient is [tex]\(x\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(x\)[/tex] by [tex]\(x^2 + x - 7\)[/tex]:
[tex]\[
x \cdot (x^2 + x - 7) = x^3 + x^2 - 7x
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
x^3 - 2x^2 - 10x + 21 - (x^3 + x^2 - 7x) = -3x^2 - 3x + 21
\][/tex]
4. Second Division Step:
- Divide the new leading term [tex]\(-3x^2\)[/tex] by the leading term [tex]\(x^2\)[/tex]:
[tex]\[
\frac{-3x^2}{x^2} = -3
\][/tex]
- The next term of the quotient is [tex]\(-3\)[/tex].
5. Multiply and Subtract:
- Multiply [tex]\(-3\)[/tex] by [tex]\(x^2 + x - 7\)[/tex]:
[tex]\[
-3 \cdot (x^2 + x - 7) = -3x^2 - 3x + 21
\][/tex]
- Subtract this from the previous result:
[tex]\[
-3x^2 - 3x + 21 - (-3x^2 - 3x + 21) = 0
\][/tex]
Since the remainder is zero, the quotient of the division is [tex]\(x - 3\)[/tex] and the remainder is zero.
Hence, the quotient is [tex]\(x - 3\)[/tex].
From the quotient, the value of [tex]\(A\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term) is [tex]\(1\)[/tex].
Therefore, the value of [tex]\(A\)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
