Use polynomial identities to factor 64x^9 − 125y^6.
Understand the Problem
The question is asking us to factor the expression 64x^9 − 125y^6 using polynomial identities. We need to identify the correct option from the provided choices.
Answer
$ (4x^3 - 5y^2)(16x^6 + 20x^3y^2 + 25y^4) $
Answer for screen readers
The factored form of the expression is:
$$ (4x^3 - 5y^2)(16x^6 + 20x^3y^2 + 25y^4) $$
Steps to Solve
- Recognize the expression type
The expression $64x^9 - 125y^6$ can be identified as a difference of cubes because it can be expressed in the form $a^3 - b^3$, where $a = (4x^3)$ and $b = (5y^2)$.
- Apply the difference of cubes formula
The difference of cubes can be factored using the identity:
$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$
Here, substitute $a$ and $b$ into the formula:
$$64x^9 - 125y^6 = (4x^3)^3 - (5y^2)^3$$
- Substitute and simplify
Now substituting into the formula gives:
$$(4x^3 - 5y^2)((4x^3)^2 + (4x^3)(5y^2) + (5y^2)^2)$$
- Calculate each term in the expanded factorization
Calculate each part:
- The first term remains $(4x^3 - 5y^2)$.
- The second term is:
- $ (4x^3)^2 = 16x^6 $
- $ (4x^3)(5y^2) = 20x^3y^2 $
- $ (5y^2)^2 = 25y^4 $
So, the second factor becomes:
$$ 16x^6 + 20x^3y^2 + 25y^4 $$
- Combine the factors
Putting it all together, we now have:
$$ 64x^9 - 125y^6 = (4x^3 - 5y^2)(16x^6 + 20x^3y^2 + 25y^4) $$
The factored form of the expression is:
$$ (4x^3 - 5y^2)(16x^6 + 20x^3y^2 + 25y^4) $$
More Information
The expression is an example of a difference of cubes, which is a common factorization in algebra. Recognizing polynomial identities is essential for simplifying expressions in mathematics.
Tips
- Confusing the difference of cubes with other polynomial identities like the sum or difference of squares.
- Forgetting to correctly apply the formula for difference of cubes, which can lead to errors in the resulting factors.
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