Question

1. which coefficients need careful handling in polynomial multiplication to avoid errors?

a. integer coefficients
b. positive coefficients
c. zero coefficients
d. negative coefficients

1. what is the product of $(x+2)$ and $(x-3)$?

a. $x^2 + x - 6$
b. $x^2 - x + 6$
c. $x^2 + 5x - 6$
d. $x^2 - x - 6$

1. what is the result of multiplying $(x+1)$ and $(x-1)$ using any method?

a. $x^2 - 1$
b. $x^2 - x + 1$
c. $2x^2 - 1$
d. $x^2 - 1$

Explanation:

Step1: Answer first multiple-choice

Negative coefficients often lead to sign errors when multiplying polynomials, as subtraction/negative signs can be mishandled during distribution.

Step2: Expand $(x+2)(x-3)$

Use FOIL method:
$x\cdot x + x\cdot(-3) + 2\cdot x + 2\cdot(-3) = x^2 -3x +2x -6 = x^2 -x -6$

Step3: Expand $(x+1)(x-1)$

Use difference of squares formula:
$(a+b)(a-b)=a^2-b^2$, so $x^2 -1^2 = x^2 -1$

Answer:

1. d. Negative coefficients
2. d. $x^2 - x - 6$
3. a. $x^2 - 1$