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

  1. 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 $b = (5y^2)$.

  1. 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:

$$64x^9 - 125y^6 = (4x^3)^3 - (5y^2)^3$$

  1. Substitute and simplify

Now substituting into the formula gives:

$$(4x^3 - 5y^2)((4x^3)^2 + (4x^3)(5y^2) + (5y^2)^2)$$

  1. 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:

$$ 16x^6 + 20x^3y^2 + 25y^4 $$

  1. 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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