Math  /  Algebra

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Given that f(x)=x23x10f(x)=x^{2}-3 x-10 and g(x)=x5g(x)=x-5, find (fg)(x)(f \cdot g)(x) and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. The function f(x)f(x) is a polynomial given by f(x)=x23x10f(x) = x^2 - 3x - 10.
2. The function g(x)g(x) is a linear polynomial given by g(x)=x5g(x) = x - 5.
3. The notation (fg)(x)(f \cdot g)(x) represents the product of f(x)f(x) and g(x)g(x).
4. To find (fg)(x)(f \cdot g)(x), we need to multiply the polynomials f(x)f(x) and g(x)g(x) and simplify the resulting expression.

STEP 2

1. Write down the expressions for f(x)f(x) and g(x)g(x).
2. Multiply the polynomials f(x)f(x) and g(x)g(x).
3. Simplify the resulting polynomial by combining like terms.

STEP 3

Write down the expressions for the given functions f(x)f(x) and g(x)g(x).
f(x)=x23x10 f(x) = x^2 - 3x - 10 g(x)=x5 g(x) = x - 5

STEP 4

Multiply the polynomials f(x)f(x) and g(x)g(x).
(fg)(x)=(x23x10)(x5) (f \cdot g)(x) = (x^2 - 3x - 10)(x - 5)

STEP 5

Distribute each term in the first polynomial by each term in the second polynomial.
(x23x10)(x5)=x2(x5)3x(x5)10(x5) (x^2 - 3x - 10)(x - 5) = x^2(x - 5) - 3x(x - 5) - 10(x - 5)

STEP 6

Multiply out each term.
x2(x5)=x35x2 x^2(x - 5) = x^3 - 5x^2 3x(x5)=3x2+15x -3x(x - 5) = -3x^2 + 15x 10(x5)=10x+50 -10(x - 5) = -10x + 50

STEP 7

Combine all the terms from the previous step.
(fg)(x)=x35x23x2+15x10x+50 (f \cdot g)(x) = x^3 - 5x^2 - 3x^2 + 15x - 10x + 50

STEP 8

Combine like terms to simplify the polynomial.
(fg)(x)=x38x2+5x+50 (f \cdot g)(x) = x^3 - 8x^2 + 5x + 50
The resulting polynomial in its simplest form is:
(fg)(x)=x38x2+5x+50 (f \cdot g)(x) = x^3 - 8x^2 + 5x + 50

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