To find the resulting polynomial for [tex]\(f(x) \cdot g(x)\)[/tex] where [tex]\( f(x) = -4x + 9 \)[/tex] and [tex]\( g(x) = 10x^2 + 6x + 3 \)[/tex], we need to follow these steps:
1. Write down the polynomials:
[tex]\[
f(x) = -4x + 9
\][/tex]
[tex]\[
g(x) = 10x^2 + 6x + 3
\][/tex]
2. Multiply [tex]\( f(x) \)[/tex] by each term of [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) \cdot g(x) = (-4x + 9) \cdot (10x^2 + 6x + 3)
\][/tex]
3. Distribute each term in [tex]\( f(x) \)[/tex] across each term in [tex]\( g(x) \)[/tex]:
- First, distribute [tex]\(-4x\)[/tex]:
[tex]\[
-4x \cdot 10x^2 = -40x^3
\][/tex]
[tex]\[
-4x \cdot 6x = -24x^2
\][/tex]
[tex]\[
-4x \cdot 3 = -12x
\][/tex]
- Next, distribute [tex]\( 9 \)[/tex]:
[tex]\[
9 \cdot 10x^2 = 90x^2
\][/tex]
[tex]\[
9 \cdot 6x = 54x
\][/tex]
[tex]\[
9 \cdot 3 = 27
\][/tex]
4. Combine all the individual products:
[tex]\[
-40x^3 - 24x^2 - 12x + 90x^2 + 54x + 27
\][/tex]
5. Combine like terms:
- The [tex]\( x^3 \)[/tex] terms:
[tex]\[
-40x^3
\][/tex]
- The [tex]\( x^2 \)[/tex] terms:
[tex]\[
-24x^2 + 90x^2 = 66x^2
\][/tex]
- The [tex]\( x \)[/tex] terms:
[tex]\[
-12x + 54x = 42x
\][/tex]
- The constant term:
[tex]\[
27
\][/tex]
6. Write the resulting polynomial in standard form:
[tex]\[
-40x^3 + 66x^2 + 42x + 27
\][/tex]
Therefore, the resulting polynomial when [tex]\( f(x) \cdot g(x) \)[/tex] is found is:
[tex]\[
-40x^3 + 66x^2 + 42x + 27
\][/tex]
