To factor the polynomial [tex]\( 15x^2 - 11x - 12 \)[/tex] completely, follow these steps:
1. Identify the polynomial and its coefficients:
The polynomial is [tex]\( 15x^2 - 11x - 12 \)[/tex]. The coefficients are:
- [tex]\( a = 15 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -11 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = -12 \)[/tex] (constant term)
2. Find two numbers that multiply to [tex]\( a \times c \)[/tex] and add to [tex]\( b \)[/tex]:
- Here, [tex]\( a \times c = 15 \times -12 = -180 \)[/tex]
- We need to find two numbers that multiply to [tex]\(-180\)[/tex] and add to [tex]\(-11\)[/tex].
These numbers are [tex]\( 9 \)[/tex] and [tex]\(-20\)[/tex] because:
- [tex]\( (9) \times (-20) = -180 \)[/tex]
- [tex]\( 9 + (-20) = -11 \)[/tex]
3. Write the middle term [tex]\( -11x \)[/tex] as the sum of these two numbers:
Rewrite [tex]\( -11x \)[/tex] using the numbers found:
[tex]\[
15x^2 + 9x - 20x - 12
\][/tex]
4. Group the terms in pairs:
[tex]\[
(15x^2 + 9x) + (-20x - 12)
\][/tex]
5. Factor out the greatest common factor (GCF) from each pair:
- From the first pair [tex]\( 15x^2 + 9x \)[/tex]:
[tex]\[
3x(5x + 3)
\][/tex]
- From the second pair [tex]\( -20x - 12 \)[/tex]:
[tex]\[
-4(5x + 3)
\][/tex]
Now we have:
[tex]\[
3x(5x + 3) - 4(5x + 3)
\][/tex]
6. Factor out the common binomial factor [tex]\( (5x + 3) \)[/tex]:
[tex]\[
(5x + 3)(3x - 4)
\][/tex]
Therefore, the factored form of [tex]\( 15x^2 - 11x - 12 \)[/tex] is [tex]\((5x + 3)(3x - 4)\)[/tex].
The correct answer is:
C. [tex]\((5x + 3)(3x - 4)\)[/tex]
