Let's factor the polynomial [tex]\( 7 x^3 + 21 x^2 - 63 x \)[/tex] step-by-step.
### Step 1: Identify the Greatest Common Factor (GCF)
First, we look for the greatest common factor of all the terms in the polynomial.
- The coefficients are 7, 21, and -63. The GCF of 7, 21, and 63 is 7.
- Each term includes [tex]\( x \)[/tex]. The smallest power of [tex]\( x \)[/tex] in the terms is [tex]\( x \)[/tex].
So, the GCF of the polynomial [tex]\( 7x^3 + 21x^2 - 63x \)[/tex] is [tex]\( 7x \)[/tex].
### Step 2: Factor out the Greatest Common Factor (GCF)
Next, we factor out [tex]\( 7x \)[/tex] from each term.
[tex]\[
7x^3 + 21x^2 - 63x = 7x(x^2) + 7x(3x) - 7x(9)
\][/tex]
Factoring out [tex]\( 7x \)[/tex], we get:
[tex]\[
7x(x^2 + 3x - 9)
\][/tex]
### Step 3: Identify the Final Factored Form
So the polynomial [tex]\( 7 x^3 + 21 x^2 - 63 x \)[/tex], when factored, is [tex]\( 7x(x^2 + 3x - 9) \)[/tex].
### Step 4: Compare with Given Options
Let's compare our result with the given options:
a. [tex]\((7x + 1)(x - 9)\)[/tex]
b. [tex]\(7x^3(3x - 9)\)[/tex]
c. [tex]\(3x(4x^2 + 7x - 21)\)[/tex]
d. [tex]\(7x(x^2 + 3x - 9)\)[/tex]
The correct choice that matches our factored form, [tex]\( 7x(x^2 + 3x - 9) \)[/tex], is:
d. [tex]\(7x(x^2 + 3x - 9)\)[/tex]
