Math  /  Algebra

QuestionLet f(x)=3x2+7f(x)=3 x^{2}+7 and g(x)=49xg(x)=4-9 x. Find (fg)(x)=f(x)g(x)(f g)(x)=f(x) g(x). (fg)(x)=27x363x(f g)(x)=-27 x^{3}-63 x (fg)(x)=27x363x+28(f g)(x)=-27 x^{3}-63 x+28 (fg)(x)=27x3+12x263x+28(f g)(x)=-27 x^{3}+12 x^{2}-63 x+28 (fg)(x)=3x29x+28(f g)(x)=3 x^{2}-9 x+28

Studdy Solution

STEP 1

1. The functions f(x)=3x2+7f(x)=3x^2+7 and g(x)=49xg(x)=4-9x are polynomial functions.
2. To find (fg)(x)(f \cdot g)(x), we need to multiply f(x)f(x) by g(x)g(x).
3. Polynomial multiplication involves distributing each term of the first polynomial to each term of the second polynomial.

STEP 2

1. Express f(x)f(x) and g(x)g(x) explicitly.
2. Perform polynomial multiplication between f(x)f(x) and g(x)g(x).
3. Simplify the resulting polynomial by combining like terms.

STEP 3

Express f(x)f(x) and g(x)g(x) explicitly:
f(x)=3x2+7 f(x) = 3x^2 + 7 g(x)=49x g(x) = 4 - 9x

STEP 4

Multiply f(x)f(x) by g(x)g(x). Distribute each term in f(x)f(x) to each term in g(x)g(x):
(fg)(x)=(3x2+7)(49x) (f g)(x) = (3x^2 + 7)(4 - 9x)

STEP 5

First, distribute 3x23x^2 to each term in 49x4 - 9x:
3x24+3x2(9x)=12x227x3 3x^2 \cdot 4 + 3x^2 \cdot (-9x) = 12x^2 - 27x^3

STEP 6

Next, distribute 77 to each term in 49x4 - 9x:
74+7(9x)=2863x 7 \cdot 4 + 7 \cdot (-9x) = 28 - 63x

STEP 7

Combine all the distributed terms:
(fg)(x)=12x227x3+2863x (f g)(x) = 12x^2 - 27x^3 + 28 - 63x

STEP 8

Arrange the polynomial in standard form (descending order of powers of xx):
(fg)(x)=27x3+12x263x+28 (f g)(x) = -27x^3 + 12x^2 - 63x + 28
Solution: (fg)(x)=27x3+12x263x+28 (f g)(x) = -27x^3 + 12x^2 - 63x + 28

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