To divide the polynomial [tex]\(P(x)\)[/tex] by the polynomial [tex]\(D(x)\)[/tex] and express [tex]\(P(x)\)[/tex] in the form of [tex]\( P(x) = D(x) \cdot Q(x) + R(x) \)[/tex], we need to perform polynomial division.
### Given Polynomials
- [tex]\( P(x) = x^3 + 16x^2 + 9x \)[/tex]
- [tex]\( D(x) = 4x^2 + 1 \)[/tex]
### Polynomial Division
Step-by-Step Solution:
1. Set Up the Division:
Divide [tex]\( P(x) = x^3 + 16x^2 + 9x \)[/tex] by [tex]\( D(x) = 4x^2 + 1 \)[/tex].
2. Determine the First Term of the Quotient:
The first term in the quotient [tex]\(Q(x)\)[/tex] is determined by dividing the leading term of the dividend [tex]\(x^3\)[/tex] by the leading term of the divisor [tex]\(4x^2\)[/tex].
[tex]\[
\frac{x^3}{4x^2} = \frac{1}{4}x = 0.25x
\][/tex]
3. Multiply and Subtract:
Multiply [tex]\(0.25x \cdot (4x^2 + 1)\)[/tex] and subtract from [tex]\(P(x)\)[/tex]:
[tex]\[
0.25x \cdot (4x^2 + 1) = x^3 + 0.25x
\][/tex]
Subtract this from [tex]\(P(x)\)[/tex]:
[tex]\[
(x^3 + 16x^2 + 9x) - (x^3 + 0.25x) = 16x^2 + 8.75x
\][/tex]
4. Repeat the Process:
Next, divide [tex]\(16x^2\)[/tex] by [tex]\(4x^2\)[/tex] to get the next term of the quotient:
[tex]\[
\frac{16x^2}{4x^2} = 4
\][/tex]
Multiply and subtract:
[tex]\[
4 \cdot (4x^2 + 1) = 16x^2 + 4
\][/tex]
Subtract from the remaining polynomial:
[tex]\[
(16x^2 + 8.75x) - (16x^2 + 4) = 8.75x - 4
\][/tex]
5. Final Quotient and Remainder:
The quotient [tex]\(Q(x)\)[/tex] is [tex]\(0.25x + 4\)[/tex], and the remainder [tex]\(R(x)\)[/tex] is [tex]\(8.75x - 4\)[/tex].
### Express [tex]\(P(x)\)[/tex] in the Required Form:
Using the results of our division, express [tex]\(P(x)\)[/tex]:
[tex]\[
P(x) = D(x) \cdot Q(x) + R(x)
\][/tex]
Substitute the quotient and remainder:
[tex]\[
x^3 + 16x^2 + 9x = (4x^2 + 1)(0.25x + 4) + (8.75x - 4)
\][/tex]
Therefore, [tex]\(P(x) = (4x^2 + 1)(0.25x + 4) + 8.75x - 4\)[/tex].
