Let's solve the given problem step-by-step.
1. Combine like terms in [tex]\( a(x) \)[/tex]:
[tex]\[
a(x) = 5x^2 - 6x + 10x - 2
\][/tex]
Combine the [tex]\( x \)[/tex] terms:
[tex]\[
a(x) = 5x^2 + 4x - 2
\][/tex]
2. Identify the degrees of [tex]\( a(x) \)[/tex] and [tex]\( b(x) \)[/tex]:
[tex]\[
\deg(a(x)) = 2 \quad \text{(since the highest power of \( x \) in \( a(x) \) is 2)}
\][/tex]
[tex]\[
\deg(b(x)) = 3 \quad \text{(since the highest power of \( x \) in \( b(x) \) is 3)}
\][/tex]
3. Perform polynomial division:
Since the degree of [tex]\( a(x) \)[/tex] is less than the degree of [tex]\( b(x) \)[/tex], the quotient [tex]\( q(x) \)[/tex] is 0.
4. Determine the remainder:
The remainder [tex]\( r(x) \)[/tex] is simply the polynomial [tex]\( a(x) \)[/tex] itself, because [tex]\( q(x) \cdot b(x) \)[/tex] would be 0 and the remaining [tex]\( a(x) \)[/tex] is unchanged.
Thus:
- The quotient [tex]\( q(x) \)[/tex]:
[tex]\[
q(x) = 0
\][/tex]
- The remainder [tex]\( r(x) \)[/tex]:
[tex]\[
r(x) = 5x^2 + 4x - 2
\][/tex]
Therefore, the final answer is:
[tex]\[
q(x) = 0
\][/tex]
[tex]\[
r(x) = 5x^2 + 4x - 2
\][/tex]
