Select the correct answer.

Which expression shows the factors of [tex][tex]$9x^2 + 3x - 2$[/tex][/tex]?

A. [tex][tex]$(3x - 1)(3x - 2)$[/tex][/tex]
B. [tex][tex]$(3x + 1)(3x - 2)$[/tex][/tex]
C. [tex][tex]$(3x + 2)(3x - 1)$[/tex][/tex]
D. [tex][tex]$(9x - 2)(x + 1)$[/tex][/tex]



Answer :

To factor the quadratic expression [tex]\( 9x^2 + 3x - 2 \)[/tex], we need to find two binomials whose product gives us this polynomial. Here are the steps to solve this problem.

1. Write down the polynomial:
[tex]\[ 9x^2 + 3x - 2 \][/tex]

2. Identify the terms in the polynomial:
- The coefficient of [tex]\( x^2 \)[/tex] is 9.
- The coefficient of [tex]\( x \)[/tex] is 3.
- The constant term is -2.

3. Find two numbers that multiply to give the product of the leading coefficient (9) and the constant term (-2).
The product of 9 and -2 is:
[tex]\[ 9 \cdot (-2) = -18 \][/tex]

4. We need two numbers that multiply to -18 and add up to the middle coefficient, 3. These numbers are 6 and -3 because:
[tex]\[ 6 \times (-3) = -18 \][/tex]
[tex]\[ 6 + (-3) = 3 \][/tex]

5. Rewrite the middle term using the two numbers found:
[tex]\[ 9x^2 + 6x - 3x - 2 \][/tex]

6. Factor by grouping:
Group the terms to factor out the common factors:
[tex]\[ (9x^2 + 6x) + (-3x - 2) \][/tex]
Factor out the greatest common factor from each group:
[tex]\[ 3x(3x + 2) - 1(3x + 2) \][/tex]

7. Factor out the common binomial factor:
Notice that [tex]\( 3x + 2 \)[/tex] is a common factor:
[tex]\[ (3x + 2)(3x - 1) \][/tex]

Therefore, the factors of the polynomial [tex]\( 9x^2 + 3x - 2 \)[/tex] are:

[tex]\[ (3x + 2)(3x - 1) \][/tex]

So, the correct answer is:

C. [tex]\( (3x + 2)(3x - 1) \)[/tex]