Question

The variables A, B, and C represent polynomials where A = x + 1, B = x2 + 2x − 1, and C = 2x. What is AB + C in simplest form?

A. x^3 + 3x − 1
B. x^3 + 4x − 1
C. x^3 + 3x^2 + 3x − 1
D. x^3 + 2x^2 − x + 1

Answer 1

AB + C is calculated by first distributing A into B and then adding C. The simplest form of AB + C is x^3 + 3x^2 + 3x - 1, corresponding with option C of the choices given.

Step-by-step explanation:

To find AB + C in simplest form, we first calculate the product of A and B, then add polynomial C. Recall that the distributive law allows us to multiply each term in the first polynomial by each term in the second polynomial. Given A = x + 1 and B = x^2 + 2x - 1, applying the distributive property:

• A * B = (x + 1) * (x^2 + 2x - 1)
• A * B = x*x^2 + x*2x + x*(-1) + 1*x^2 + 1*2x + 1*(-1)
• A * B = x^3 + 2x^2 - x + x^2 + 2x - 1
• A * B = x^3 + 3x^2 + x - 1

Now we add polynomial C, which is 2x, to this result:

• AB + C = (x^3 + 3x^2 + x - 1) + 2x
• AB + C = x^3 + 3x^2 + 3x - 1

Therefore, AB + C in simplest form is x^3 + 3x^2 + 3x - 1, which corresponds with option C of the multiple choice question.