Question㪱, Factor the polynomial. 㸚 Ald factors in your answer should have integer coefficients. $135 p^{3}+5000 q^{3}=$ $\square$ Submit

Studdy Solution

STEP 1

1. We are given a polynomial expression $135 p^{3} + 5000 q^{3}$ .
2. The task is to factor this polynomial.
3. All factors should have integer coefficients.

STEP 2

1. Identify and factor out the greatest common factor (GCF) from the polynomial.
2. Recognize the sum of cubes and apply the sum of cubes formula.

STEP 3

Identify the greatest common factor (GCF) of the coefficients 135 and 5000.
The GCF of 135 and 5000 is 5.
Factor out the GCF from the polynomial:
$135 p^{3} + 5000 q^{3} = 5(27 p^{3} + 1000 q^{3})$

STEP 4

Recognize the expression inside the parentheses as a sum of cubes:
$27 p^{3} + 1000 q^{3}$
This can be written as:
$(3p)^{3} + (10q)^{3}$
Apply the sum of cubes formula:
$a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})$
Here, $a = 3p$ and $b = 10q$ .
Substitute into the formula:
$(3p + 10q)((3p)^{2} - (3p)(10q) + (10q)^{2})$

STEP 5

Simplify the expression:
1. Calculate $(3p)^{2} = 9p^{2}$ .
2. Calculate $(3p)(10q) = 30pq$ .
3. Calculate $(10q)^{2} = 100q^{2}$ .
Substitute these into the expression:
$(3p + 10q)(9p^{2} - 30pq + 100q^{2})$
Combine with the GCF factored out earlier:
$5(3p + 10q)(9p^{2} - 30pq + 100q^{2})$
The fully factored form of the polynomial is:
$\boxed{5(3p + 10q)(9p^{2} - 30pq + 100q^{2})}$

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