Sure, let's go through the steps to factor the polynomial [tex]\(2x^3 + 5x^2 - 8x - 20\)[/tex] completely.
1. Identify the Polynomial and Look for Common Factors:
First, check if there's a common factor in all the terms. In this case, there isn't a common factor besides 1.
2. Factor by Grouping:
Group the terms in pairs and factor out the common factor in each pair.
[tex]\[
2x^3 + 5x^2 - 8x - 20 = (2x^3 + 5x^2) + (-8x - 20)
\][/tex]
3. Factor Each Group:
Factor out the greatest common divisor from each group.
[tex]\[
2x^2(x + \frac{5}{2}) - 4(2x + 5)
\][/tex]
Note that here we recognized [tex]\(2x^2\)[/tex] from the first group and [tex]\(4\)[/tex] from the second group. However, upon revisiting the original equation, it might be clearer if we factor entirely differently:
[tex]\[
x^2(2x + 5) - 4(2x + 5)
\][/tex]
Then we notice a common binomial factor [tex]\((2x + 5)\)[/tex]:
[tex]\[
(x^2 - 4)(2x + 5)
\][/tex]
4. Further Factor Remaining Quadratic Expression:
Notice that [tex]\(x^2 - 4\)[/tex] is a difference of squares:
[tex]\[
x^2 - 4 = (x - 2)(x + 2)
\][/tex]
5. Combine All Factors:
Combine the factors obtained:
[tex]\[
2x^3 + 5x^2 - 8x - 20 = (x - 2)(x + 2)(2x + 5)
\][/tex]
Thus, the completely factored form of the polynomial [tex]\(2x^3 + 5x^2 - 8x - 20\)[/tex] is:
[tex]\[
(x - 2)(x + 2)(2x + 5)
\][/tex]
