Given the polynomials [tex]\(a(x) = -12x^5 - 2x^3 - 9x\)[/tex] and [tex]\(b(x) = 3x^4 + x^2 + 1\)[/tex], we want to perform polynomial division to find the quotient [tex]\(q(x)\)[/tex] and the remainder [tex]\(r(x)\)[/tex].
To divide [tex]\(a(x)\)[/tex] by [tex]\(b(x)\)[/tex]:
1. Setup the division: Divide the leading term of [tex]\(a(x)\)[/tex] by the leading term of [tex]\(b(x)\)[/tex]:
[tex]\[
\frac{-12x^5}{3x^4} = -4x
\][/tex]
2. Form the first term of quotient: The term [tex]\(-4x\)[/tex] becomes the first term of the quotient [tex]\(q(x)\)[/tex].
3. Multiply and subtract: Multiply [tex]\(b(x)\)[/tex] by the first term of the quotient:
[tex]\[
-4x \cdot (3x^4 + x^2 + 1) = -12x^5 - 4x^3 - 4x
\][/tex]
Subtract this from [tex]\(a(x)\)[/tex]:
[tex]\[
(-12x^5 - 2x^3 - 9x) - (-12x^5 - 4x^3 - 4x) = 2x^3 - 5x
\][/tex]
4. Form the remainder: The new polynomial after subtraction, [tex]\(2x^3 - 5x\)[/tex], is of degree less than [tex]\(b(x)\)[/tex], so it is the remainder [tex]\(r(x)\)[/tex].
Hence, we have the quotient and remainder as:
[tex]\[
q(x) = -4x
\][/tex]
[tex]\[
r(x) = 2x^3 - 5x
\][/tex]
So, the quotient [tex]\(q(x)\)[/tex] is [tex]\(-4x\)[/tex] and the remainder [tex]\(r(x)\)[/tex] is [tex]\(2x^3 - 5x\)[/tex].
