A company's weekly revenue, R(x), and weekly cost, C(x), in dollars, are modeled by the fu… — ACT

A company's weekly revenue, $R(x)$ , and weekly cost, $C(x)$ , in dollars, are modeled by the functions $R(x) = (2x^2 + 3)^2$ and $C(x) = x^2(x^3 - 5)$ , where $x$ represents the number of units produced and sold. Which of the following expressions represents the company's weekly profit, $P(x) = R(x) - C(x)$ ?

A
$-x^5 + 4x^4 + 7x^2 + 9$
B
$-x^6 + 4x^4 + 17x^2 + 9$
$-x^5 + 4x^4 + 17x^2 + 9$ Cevap
D
$-x^5 + 4x^4 + 5x^2 + 9$
E
$-x^6 + 4x^4 + 7x^2 + 9$

Cevap

The expression representing the weekly profit is

-x^5 + 4x^4 + 17x^2 + 9

To find the profit function, we subtract the cost function from the revenue function. First, expanding

(2x^2 + 3)^2

yields

4x^4 + 12x^2 + 9

. Second, distributing

x^2

over

(x^3 - 5)

yields

x^5 - 5x^2

. Subtracting these two functions gives

(4x^4 + 12x^2 + 9) - (x^5 - 5x^2)

. Distributing the subtraction sign changes the signs of the cost function to

-x^5 + 5x^2

. Combining like terms (

12x^2 + 5x^2 = 17x^2

) and organizing the expression in descending order yields

-x^5 + 4x^4 + 17x^2 + 9

Adım Adım Çözüm

Expand the revenue function

R(x) = (2x^2 + 3)^2

using the binomial square identity

(a+b)^2 = a^2 + 2ab + b^2

R(x) = (2x^2)^2 + 2(2x^2)(3) + 3^2 = 4x^4 + 12x^2 + 9

This simplifies the revenue expression into a standard polynomial form.

Expand the cost function

C(x) = x^2(x^3 - 5)

by distributing

x^2

to both terms inside the parentheses and applying the product rule for exponents

x^a \cdot x^b = x^{a+b}

C(x) = x^2(x^3) - x^2(5) = x^5 - 5x^2

This simplifies the cost expression into a standard polynomial form.

Subtract

C(x)

from

R(x)

by setting up

P(x) = R(x) - C(x)

and distributing the negative sign to all terms of

C(x)

P(x) = (4x^4 + 12x^2 + 9) - (x^5 - 5x^2) = 4x^4 + 12x^2 + 9 - x^5 + 5x^2

Profit is revenue minus cost, and distributing the negative sign correctly is essential.

Combine like terms and write the final expression in descending order of exponents.

P(x) = -x^5 + 4x^4 + 17x^2 + 9

To present the polynomial in standard form.

Anahtar Kavram

Polynomial operations, including expanding binomial squares, distributing monomial terms, and subtracting polynomials with careful attention to sign distribution.

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