To divide the polynomial [tex]\( P(x) \)[/tex] by [tex]\( D(x) \)[/tex] using polynomial long division, we follow these steps:
Given:
[tex]\[ P(x) = 16x^3 + 9x^2 \][/tex]
[tex]\[ D(x) = 20x^4 + 4x^2 + 1 \][/tex]
We want to express [tex]\( P(x) \)[/tex] as:
[tex]\[ P(x) = D(x) \cdot Q(x) + R(x) \][/tex]
where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R(x) \)[/tex] is the remainder.
1. Set up the division:
- [tex]\( P(x) = 16x^3 + 9x^2 \)[/tex]
- [tex]\( D(x) = 20x^4 + 4x^2 + 1 \)[/tex]
2. Determine the leading term of the quotient:
- Compare the degrees of [tex]\( P(x) \)[/tex] and [tex]\( D(x) \)[/tex].
- The degree of [tex]\( P(x) \)[/tex] is 3 (due to [tex]\( 16x^3 \)[/tex]).
- The degree of [tex]\( D(x) \)[/tex] is 4 (due to [tex]\( 20x^4 \)[/tex]).
Since the degree of [tex]\( P(x) \)[/tex] is less than the degree of [tex]\( D(x) \)[/tex], the quotient [tex]\( Q(x) \)[/tex] will be 0.
3. Division process:
- Given that [tex]\( P(x) \)[/tex] has a lower degree than [tex]\( D(x) \)[/tex], [tex]\( D(x) \)[/tex] cannot divide [tex]\( P(x) \)[/tex]. Therefore, the quotient [tex]\( Q(x) \)[/tex] is 0.
- The entire polynomial [tex]\( P(x) \)[/tex] becomes the remainder [tex]\( R(x) \)[/tex].
4. Final expression:
[tex]\[ P(x) = (20x^4 + 4x^2 + 1) \cdot 0 + 16x^3 + 9x^2 \][/tex]
[tex]\[ P(x) = 0 + 16x^3 + 9x^2 \][/tex]
Therefore, the quotient [tex]\( Q(x) \)[/tex] is [tex]\( 0 \)[/tex] and the remainder [tex]\( R(x) \)[/tex] is [tex]\( 16x^3 + 9x^2 \)[/tex].
So, the answer is:
[tex]\[ P(x) = 0 + 16x^3 + 9x^2 \][/tex]
This means:
[tex]\[ Q(x) = 0 \][/tex]
[tex]\[ R(x) = 16x^3 + 9x^2 \][/tex]
