Question

the function ( j(x) = f(x) - g(x) ) is the difference of two polynomials.

( j(x) = (2x^3 + 5x^2 + 2) - (-2x^3 - 4x^2 + 3x) )

a. discuss: what do you think the graph of ( j(x) ) looks like?

b. write ( j(x) ) as a sum using the fewest number of terms.

( j(x) = )

Explanation:

Step1: Distribute the negative sign

$j(x) = 2x^3 + 5x^2 + 2 + 2x^3 + 4x^2 - 3x$

Step2: Combine like terms (cubic terms)

$2x^3 + 2x^3 = 4x^3$

Step3: Combine like terms (quadratic terms)

$5x^2 + 4x^2 = 9x^2$

Step4: Keep linear and constant terms

$-3x + 2$

Step5: Combine all simplified terms

$j(x) = 4x^3 + 9x^2 - 3x + 2$

Step6: Describe the graph shape

As a cubic with positive leading coefficient, the graph falls to the left, rises to the right, and has at most 2 turning points.

Answer:

a. The graph of $j(x)$ is a cubic polynomial curve: it falls as $x \to -\infty$, rises as $x \to +\infty$, and can have up to 2 turning points (local maximum and minimum).
b. $j(x) = 4x^3 + 9x^2 - 3x + 2$