To factor the polynomial [tex]\(64x^3 y + 27 y^4\)[/tex] with integer coefficients, follow these steps:
1. Identify the common factor:
- Both terms in the polynomial, [tex]\(64x^3 y\)[/tex] and [tex]\(27y^4\)[/tex], have [tex]\(y\)[/tex] as a common factor.
- Factor out [tex]\(y\)[/tex] from the polynomial:
[tex]\[
64x^3 y + 27 y^4 = y (64x^3 + 27 y^3)
\][/tex]
2. Recognize the polynomial form:
- The expression [tex]\(64x^3 + 27y^3\)[/tex] is a sum of cubes.
- There is a formula for factoring a sum of cubes: [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex].
3. Apply the sum of cubes formula:
- Let [tex]\(a = 4x\)[/tex] and [tex]\(b = 3y\)[/tex] since [tex]\((4x)^3 = 64x^3\)[/tex] and [tex]\((3y)^3 = 27y^3\)[/tex].
- Using the formula, we write:
[tex]\[
64x^3 + 27y^3 = (4x + 3y)((4x)^2 - (4x)(3y) + (3y)^2)
\][/tex]
4. Simplify the terms:
- Calculate the squares and product within the factors:
[tex]\[
(4x)^2 = 16x^2
\][/tex]
[tex]\[
(4x)(3y) = 12xy
\][/tex]
[tex]\[
(3y)^2 = 9y^2
\][/tex]
- Combine these results:
[tex]\[
64x^3 + 27y^3 = (4x + 3y)(16x^2 - 12xy + 9y^2)
\][/tex]
5. Include the common factor:
- Combine the final factored expression with the common factor [tex]\(y\)[/tex]:
[tex]\[
64x^3 y + 27 y^4 = y(4x + 3y)(16x^2 - 12xy + 9y^2)
\][/tex]
Therefore, the fully factored form of the polynomial [tex]\(64x^3 y + 27 y^4\)[/tex] is:
[tex]\[
y(4x + 3y)(16x^2 - 12xy + 9y^2)
\][/tex]
