To solve the problem of finding [tex]\((f \cdot g)(x)\)[/tex] and [tex]\((f \div g)(x)\)[/tex] for the given functions [tex]\(f(x) = 15x^2 + 19x + 6\)[/tex] and [tex]\(g(x) = 5x + 3\)[/tex], let's go through the process step-by-step.
### Multiplying the Polynomials
To find [tex]\((f \cdot g)(x)\)[/tex], we multiply the two polynomials:
[tex]\[ f(x) = 15x^2 + 19x + 6 \][/tex]
[tex]\[ g(x) = 5x + 3 \][/tex]
We distribute each term in [tex]\(g(x)\)[/tex] across the polynomial [tex]\(f(x)\)[/tex]:
[tex]\[
(15x^2 + 19x + 6) \cdot (5x + 3)
\][/tex]
First, distribute [tex]\(5x\)[/tex] across each term in [tex]\(f(x)\)[/tex]:
[tex]\[
5x \cdot 15x^2 = 75x^3
\][/tex]
[tex]\[
5x \cdot 19x = 95x^2
\][/tex]
[tex]\[
5x \cdot 6 = 30x
\][/tex]
Next, distribute [tex]\(3\)[/tex] across each term in [tex]\(f(x)\)[/tex]:
[tex]\[
3 \cdot 15x^2 = 45x^2
\][/tex]
[tex]\[
3 \cdot 19x = 57x
\][/tex]
[tex]\[
3 \cdot 6 = 18
\][/tex]
Now, add all these products together:
[tex]\[
75x^3 + 95x^2 + 30x + 45x^2 + 57x + 18
\][/tex]
Combine like terms:
[tex]\[
75x^3 + (95x^2 + 45x^2) + (30x + 57x) + 18
\][/tex]
[tex]\[
75x^3 + 140x^2 + 87x + 18
\][/tex]
Thus, [tex]\((f \cdot g)(x) = 75x^3 + 140x^2 + 87x + 18\)[/tex].
### Dividing the Polynomials
To find [tex]\((f \div g)(x)\)[/tex], we divide [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex]:
[tex]\[ f(x) = 15x^2 + 19x + 6 \][/tex]
[tex]\[ g(x) = 5x + 3 \][/tex]
Perform polynomial division:
We start by dividing the leading term of [tex]\(f(x)\)[/tex] by the leading term of [tex]\(g(x)\)[/tex]:
[tex]\[
\frac{15x^2}{5x} = 3x
\][/tex]
Now, multiply [tex]\(g(x)\)[/tex] by [tex]\(3x\)[/tex] and subtract this product from [tex]\(f(x)\)[/tex]:
[tex]\[
(15x^2 + 19x + 6) - (3x \cdot (5x + 3))
\][/tex]
[tex]\[
(15x^2 + 19x + 6) - (15x^2 + 9x)
\][/tex]
[tex]\[
(15x^2 + 19x + 6) - 15x^2 - 9x
\][/tex]
[tex]\[
(15x^2 - 15x^2) + (19x - 9x) + 6
\][/tex]
[tex]\[
10x + 6
\][/tex]
Next, divide [tex]\(10x\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[
\frac{10x}{5x} = 2
\][/tex]
Multiply [tex]\(g(x)\)[/tex] by [tex]\(2\)[/tex] and subtract from the remaining polynomial:
[tex]\[
(10x + 6) - (2 \cdot (5x + 3))
\][/tex]
[tex]\[
(10x + 6) - (10x + 6)
\][/tex]
[tex]\[
10x + 6 - 10x - 6
\][/tex]
[tex]\[
0
\][/tex]
So, the quotient is:
[tex]\[
\boxed{3x + 2}
\][/tex]
Putting everything together, we have:
[tex]\[
(f \cdot g)(x) = 75x^3 + 140x^2 + 87x + 18
\][/tex]
[tex]\[
(f \div g)(x) = 3x + 2
\][/tex]
Thus, the correct answers are:
[tex]\((f \cdot g)(x) = 75x^3 + 140x^2 + 87x + 18\)[/tex]
[tex]\((f \div g)(x) = 3x + 2\)[/tex]
