Answer :

Sure, I'd be happy to help you solve the polynomial equation [tex]\(2x^3 - 12x^2 - 8x + 48 = 0\)[/tex] by grouping. Here's a detailed, step-by-step solution:

1. Group the terms in pairs:
[tex]\[
(2x^3 - 12x^2) + (-8x + 48) = 0
\][/tex]

2. Factor out the greatest common factor (GCD) from each group:
[tex]\[
2x^2(x - 6) - 8(x - 6) = 0
\][/tex]

3. Factor out the common binomial factor [tex]\((x - 6)\)[/tex]:
[tex]\[
(2x^2 - 8)(x - 6) = 0
\][/tex]

4. Simplify the expression inside the parentheses:
[tex]\[
2(x^2 - 4)(x - 6) = 0
\][/tex]

5. Recognize that [tex]\(x^2 - 4\)[/tex] is a difference of squares and factor it further:
[tex]\[
2(x - 2)(x + 2)(x - 6) = 0
\][/tex]

6. Set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
[tex]\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\][/tex]

Thus, the solutions to the equation [tex]\(2x^3 - 12x^2 - 8x + 48 = 0\)[/tex] are:
[tex]\[
x = -2, \; x = 2, \; x = 6
\][/tex]

Thanks for taking the time to read Solve the following polynomial equation by grouping tex 2x 3 12x 2 8x 48 0 tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada