To solve the problem of dividing the polynomial [tex]\(a(x) = -13x^9 + 6x^5 + 12x\)[/tex] by the polynomial [tex]\(b(x) = x^6\)[/tex], we will find the quotient polynomial [tex]\(q(x)\)[/tex] and the remainder polynomial [tex]\(r(x)\)[/tex].
Step-by-Step Solution:
1. Identify the degrees:
- The degree of [tex]\(a(x)\)[/tex] is [tex]\(9\)[/tex] (the highest exponent in [tex]\(a(x)\)[/tex]).
- The degree of [tex]\(b(x)\)[/tex] is [tex]\(6\)[/tex] (the highest exponent in [tex]\(b(x)\)[/tex]).
2. Perform Polynomial Division:
- To divide, we look at the leading terms of [tex]\(a(x)\)[/tex] and [tex]\(b(x)\)[/tex].
- The leading term of [tex]\(a(x)\)[/tex] is [tex]\(-13x^9\)[/tex].
- The leading term of [tex]\(b(x)\)[/tex] is [tex]\(x^6\)[/tex].
3. Divide the leading term of [tex]\(a(x)\)[/tex] by the leading term of [tex]\(b(x)\)[/tex]:
- [tex]\(\frac{-13x^9}{x^6} = -13x^3\)[/tex].
4. Subtract:
- Multiply [tex]\(b(x) = x^6\)[/tex] by [tex]\(-13x^3\)[/tex] to get [tex]\(-13x^9\)[/tex].
- Subtract this product from [tex]\(a(x)\)[/tex]:
[tex]\[
(-13x^9 + 6x^5 + 12x) - (-13x^9) = 6x^5 + 12x
\][/tex]
5. Repeat the process if necessary:
- Now the new polynomial we have is [tex]\(6x^5 + 12x\)[/tex].
- Since the degree of this polynomial is less than the degree of [tex]\(b(x) = x^6\)[/tex], we can stop here.
6. Find the quotient and remainder:
- The quotient polynomial [tex]\(q(x)\)[/tex] is obtained from the division process: [tex]\(q(x) = -13x^3\)[/tex].
- The remainder polynomial [tex]\(r(x)\)[/tex] is the polynomial left after subtraction: [tex]\(r(x) = 6x^5 + 12x\)[/tex].
Therefore, when dividing [tex]\(a(x) = -13x^9 + 6x^5 + 12x\)[/tex] by [tex]\(b(x) = x^6\)[/tex]:
The quotient, [tex]\(q(x)\)[/tex], is:
[tex]\[
q(x) = -13x^3
\][/tex]
The remainder, [tex]\(r(x)\)[/tex], is:
[tex]\[
r(x) = 6x^5 + 12x
\][/tex]
