Question

use the functions to answer the question.
f(x) = 2x² + 5x - 12
g(x) = 2x - 3
choose all arithmetic combinations that are correct.
a. f(x) + g(x) = 2x² + 7x - 15
b. f(x) - g(x) = 5x - 9
c. f(x) × g(x) = 3x³ - 5x - 12
d. f(x) ÷ g(x) = x + 4

Explanation:

Step1: Check Option A (Addition)

To find \( f(x) + g(x) \), we add the two functions:
\( f(x) + g(x) = (2x^2 + 5x - 12) + (2x - 3) \)
Combine like terms: \( 2x^2 + (5x + 2x) + (-12 - 3) = 2x^2 + 7x - 15 \). So Option A is correct.

Step2: Check Option B (Subtraction)

\( f(x) - g(x) = (2x^2 + 5x - 12) - (2x - 3) \)
Distribute the negative sign: \( 2x^2 + 5x - 12 - 2x + 3 \)
Combine like terms: \( 2x^2 + 3x - 9 \). This is not equal to \( 5x - 9 \), so Option B is incorrect.

Step3: Check Option C (Multiplication)

\( f(x) \times g(x) = (2x^2 + 5x - 12)(2x - 3) \)
Use the distributive property (FOIL for polynomials):
\( 2x^2(2x) + 2x^2(-3) + 5x(2x) + 5x(-3) - 12(2x) - 12(-3) \)
\( = 4x^3 - 6x^2 + 10x^2 - 15x - 24x + 36 \)
Combine like terms: \( 4x^3 + 4x^2 - 39x + 36 \). This is not equal to \( 3x^3 - 5x - 12 \), so Option C is incorrect.

Step4: Check Option D (Division)

We can perform polynomial long division or factor \( f(x) \). Let's factor \( f(x) = 2x^2 + 5x - 12 \). We need two numbers that multiply to \( 2 \times (-12) = -24 \) and add to 5. Those numbers are 8 and -3. So,
\( 2x^2 + 8x - 3x - 12 = 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4) \)
Now, \( \frac{f(x)}{g(x)} = \frac{(2x - 3)(x + 4)}{2x - 3} \) (assuming \( 2x - 3
eq 0 \)), which simplifies to \( x + 4 \). So Option D is correct.

Answer:

A. \( f(x) + g(x) = 2x^2 + 7x - 15 \)
D. \( f(x) \div g(x) = x + 4 \)