To solve the polynomial equation [tex]\( 7x^3 + 21x^2 - 63x = 0 \)[/tex], follow these steps:
1. Factor out the greatest common factor:
The polynomial [tex]\( 7x^3 + 21x^2 - 63x \)[/tex] has a common factor of [tex]\( 7x \)[/tex]. Factoring [tex]\( 7x \)[/tex] out, we get:
[tex]\[
7x (x^2 + 3x - 9) = 0
\][/tex]
2. Set each factor to zero:
For a product to be zero, at least one of the factors must be zero. Therefore, set each factor to zero and solve separately:
[tex]\[
7x = 0 \quad \text{or} \quad x^2 + 3x - 9 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex] from the first factor:
Solving [tex]\( 7x = 0 \)[/tex]:
[tex]\[
x = 0
\][/tex]
4. Solve the quadratic equation:
Now we solve the quadratic equation [tex]\( x^2 + 3x - 9 = 0 \)[/tex]. We use the quadratic formula, [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -9 \)[/tex]:
[tex]\[
x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-9)}}{2(1)}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-3 \pm \sqrt{9 + 36}}{2}
\][/tex]
[tex]\[
x = \frac{-3 \pm \sqrt{45}}{2}
\][/tex]
[tex]\[
x = \frac{-3 \pm 3\sqrt{5}}{2}
\][/tex]
The solutions to the quadratic equation are:
[tex]\[
x = \frac{-3 + 3\sqrt{5}}{2} \quad \text{and} \quad x = \frac{-3 - 3\sqrt{5}}{2}
\][/tex]
5. Combine all solutions:
Therefore, the solutions to the original polynomial equation are:
[tex]\[
x = 0, \quad x = \frac{-3 + 3\sqrt{5}}{2}, \quad x = \frac{-3 - 3\sqrt{5}}{2}
\][/tex]
By comparing with the answer choices provided:
- a. [tex]\( 0, \frac{-3 \pm 3\sqrt{5}}{2} \)[/tex]
- b. [tex]\( -9, 3, 7 \)[/tex]
- c. [tex]\( \frac{-3 \pm 3\sqrt{5}}{2} \)[/tex]
- d. [tex]\( 0, \frac{3 \pm \sqrt{5}}{2} \)[/tex]
The correct answer is:
a. [tex]\( 0, \frac{-3 \pm 3\sqrt{5}}{2} \)[/tex]
