Sure, let's analyze the problem step-by-step.
1. We start by considering the given polynomials:
- [tex]\( p(x) \)[/tex] is a polynomial of degree 1. We can generally represent this as [tex]\( p(x) = ax + b \)[/tex] where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
- [tex]\( q(x) \)[/tex] is a polynomial of degree 2. We can generally represent this as [tex]\( q(x) = cx^2 + dx + e \)[/tex] where [tex]\( c \)[/tex], [tex]\( d \)[/tex], and [tex]\( e \)[/tex] are constants.
2. To find the product [tex]\( p(x) \times q(x) \)[/tex], we multiply these polynomials together:
[tex]\[
p(x) \times q(x) = (ax + b) \times (cx^2 + dx + e)
\][/tex]
3. We expand this product by distributing each term in [tex]\( p(x) \)[/tex] across each term in [tex]\( q(x) \)[/tex]:
[tex]\[
(ax + b) \times (cx^2 + dx + e) = ax \cdot cx^2 + ax \cdot dx + ax \cdot e + b \cdot cx^2 + b \cdot dx + b \cdot e
\][/tex]
4. Simplify this by combining like terms:
[tex]\[
= acx^3 + adx^2 + aex + bcx^2 + bdx + be
\][/tex]
5. Combining the terms, we organize them by the powers of [tex]\( x \)[/tex]:
[tex]\[
= acx^3 + (ad + bc)x^2 + (ae + bd)x + be
\][/tex]
6. The highest degree term in the expanded polynomial is [tex]\( acx^3 \)[/tex], which indicates that the leading term has an exponent of 3.
Thus, the result of multiplying [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex] is a polynomial of degree 3.
The key points summarized:
- [tex]\( p(x) \)[/tex] has a degree of 1.
- [tex]\( q(x) \)[/tex] has a degree of 2.
- The product [tex]\( p(x) \times q(x) \)[/tex] has a degree of [tex]\( 1 + 2 = 3 \)[/tex].
Therefore, the polynomial [tex]\( p(x) \times q(x) \)[/tex] is a polynomial of degree 3.
