To find the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] such that [tex]\( 3x^3 + 6x^2 - 24x \)[/tex] is factored in the form [tex]\( a x(x + b)(x + c) \)[/tex], we can follow these steps:
1. Factor out the common term:
The given polynomial is [tex]\( 3x^3 + 6x^2 - 24x \)[/tex]. The common factor in all the terms is [tex]\( 3x \)[/tex].
[tex]\[
3x^3 + 6x^2 - 24x = 3x(x^2 + 2x - 8)
\][/tex]
2. Factor the quadratic expression:
Now we need to factor the quadratic polynomial [tex]\( x^2 + 2x - 8 \)[/tex].
To do this, we look for two numbers that multiply to [tex]\(-8\)[/tex] (the constant term) and add to [tex]\(2\)[/tex] (the coefficient of [tex]\(x\)[/tex]). These two numbers are [tex]\(4\)[/tex] and [tex]\(-2\)[/tex], because:
[tex]\( 4 \times -2 = -8 \)[/tex]
[tex]\( 4 + (-2) = 2 \)[/tex]
So, we can write:
[tex]\[
x^2 + 2x - 8 = (x + 4)(x - 2)
\][/tex]
3. Combine the factors:
Using the factors from the previous steps, we get:
[tex]\[
3x(x^2 + 2x - 8) = 3x(x + 4)(x - 2)
\][/tex]
Now, we can identify the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -2 \)[/tex]
So, the missing values are:
[tex]\[ a = 3, \, b = 4, \, c = -2 \][/tex]
