To factor the polynomial [tex]\( 5p^3 + 40 \)[/tex], follow these steps:
1. Factor out the greatest common factor (GCF):
First, identify the greatest common factor of all the terms in the polynomial.
In this case, the terms are [tex]\(5p^3\)[/tex] and 40, and the GCF is 5.
[tex]\[
5p^3 + 40 = 5(p^3 + 8)
\][/tex]
2. Recognize the sum of cubes:
Next, observe that [tex]\(p^3 + 8\)[/tex] can be written as a sum of cubes. Recall the sum of cubes formula:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
Here, [tex]\(p^3 + 8\)[/tex] can be expressed as [tex]\(p^3 + 2^3\)[/tex].
3. Apply the sum of cubes formula:
Substitute [tex]\(a = p\)[/tex] and [tex]\(b = 2\)[/tex] into the sum of cubes formula:
[tex]\[
p^3 + 2^3 = (p + 2)(p^2 - 2p + 4)
\][/tex]
4. Combine the factored terms:
Now, combine the result with the GCF factored out earlier:
[tex]\[
5(p^3 + 8) = 5(p + 2)(p^2 - 2p + 4)
\][/tex]
Thus, the factored form of the polynomial [tex]\( 5p^3 + 40 \)[/tex] with integer coefficients is:
[tex]\[
\boxed{5(p + 2)(p^2 - 2p + 4)}
\][/tex]
