Factor 27y³ + 8 completely.
Understand the Problem
The question is asking how to factor the expression 27y³ + 8 completely. This involves recognizing that the expression is a sum of cubes and applying the sum of cubes factoring formula.
Answer
The complete factorization is $27y^3 + 8 = (3y + 2)(9y^2 - 6y + 4)$.
Answer for screen readers
The complete factorization of the expression is: $$ 27y^3 + 8 = (3y + 2)(9y^2 - 6y + 4) $$
Steps to Solve
- Identify the expression as a sum of cubes
Recognize that the expression $27y^3 + 8$ can be written as a sum of cubes, where $27y^3$ is $(3y)^3$ and $8$ is $2^3$.
- Apply the sum of cubes formula
The sum of cubes formula is given by: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ Here, let $a = 3y$ and $b = 2$.
- Substitute values into the formula
Substituting $a = 3y$ and $b = 2$ into the sum of cubes formula: $$ 27y^3 + 8 = (3y + 2)((3y)^2 - (3y)(2) + 2^2) $$
- Simplify the expression
Now calculate each term in the second factor:
- Calculate $(3y)^2$: $$ (3y)^2 = 9y^2 $$
- Calculate $-(3y)(2)$: $$ -(3y)(2) = -6y $$
- Calculate $2^2$: $$ 2^2 = 4 $$
Combine these terms: $$ 9y^2 - 6y + 4 $$
Therefore, the factorization is: $$ 27y^3 + 8 = (3y + 2)(9y^2 - 6y + 4) $$
The complete factorization of the expression is: $$ 27y^3 + 8 = (3y + 2)(9y^2 - 6y + 4) $$
More Information
The sum of cubes formula is a fundamental concept in algebra that facilitates the factorization of cubic expressions. Recognizing expressions as sums or differences of cubes can simplify many polynomial problems.
Tips
- Not recognizing the sum of cubes: Some might overlook that the expression can be factored as a sum of cubes. It's essential to identify both terms as perfect cubes.
- Incorrectly applying the formula: Ensure that you substitute correctly into the formula, especially when calculating $a^2$, $-ab$, and $b^2$.
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