To determine the resulting polynomial when multiplying [tex]\( f(x) = 10x + 3 \)[/tex] and [tex]\( g(x) = 10x^2 + 10x + 5 \)[/tex], we need to distribute each term in [tex]\( f(x) \)[/tex] to each term in [tex]\( g(x) \)[/tex] and combine like terms.
### Step-by-Step Solution:
1. Distribute [tex]\( 10x \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
10x \cdot (10x^2 + 10x + 5) = 10x \cdot 10x^2 + 10x \cdot 10x + 10x \cdot 5
\][/tex]
- [tex]\( 10x \cdot 10x^2 = 100x^3 \)[/tex]
- [tex]\( 10x \cdot 10x = 100x^2 \)[/tex]
- [tex]\( 10x \cdot 5 = 50x \)[/tex]
Combining these, we get:
[tex]\[
100x^3 + 100x^2 + 50x
\][/tex]
2. Distribute [tex]\( 3 \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[
3 \cdot (10x^2 + 10x + 5) = 3 \cdot 10x^2 + 3 \cdot 10x + 3 \cdot 5
\][/tex]
- [tex]\( 3 \cdot 10x^2 = 30x^2 \)[/tex]
- [tex]\( 3 \cdot 10x = 30x \)[/tex]
- [tex]\( 3 \cdot 5 = 15 \)[/tex]
Combining these, we get:
[tex]\[
30x^2 + 30x + 15
\][/tex]
3. Combine all the terms obtained from each distribution:
[tex]\[
100x^3 + 100x^2 + 50x + 30x^2 + 30x + 15
\][/tex]
Now, combine like terms:
- Combine [tex]\( x^2 \)[/tex] terms: [tex]\( 100x^2 + 30x^2 = 130x^2 \)[/tex]
- Combine [tex]\( x \)[/tex] terms: [tex]\( 50x + 30x = 80x \)[/tex]
So, the resulting polynomial is:
[tex]\[
100x^3 + 130x^2 + 80x + 15
\][/tex]
Thus, the product of [tex]\( f(x) = 10x + 3 \)[/tex] and [tex]\( g(x) = 10x^2 + 10x + 5 \)[/tex] is:
[tex]\[
f(x) \cdot g(x) = 100x^3 + 130x^2 + 80x + 15
\][/tex]
