Factor each polynomial. See Examples 5 and 6. $27z^9+64y^{12}$

Verified step by step guidance

1. Recognize that the polynomial \$27z^9 + 64y^{12}\( is a sum of two cubes because \)27z^9 = (3z^3)^3\( and \)64y^{12} = (4y^4)^3$.
2. Recall the sum of cubes factoring formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
3. Identify \(a = 3z^3\) and \(b = 4y^4\) in the expression \$27z^9 + 64y^{12}$.
4. Apply the formula by substituting \(a\) and \(b\): write the factorization as \((3z^3 + 4y^4)((3z^3)^2 - (3z^3)(4y^4) + (4y^4)^2)\).
5. Simplify each term inside the second factor: \((3z^3)^2 = 9z^6\), \((3z^3)(4y^4) = 12z^3y^4\), and \((4y^4)^2 = 16y^8\), so the full factorization is \((3z^3 + 4y^4)(9z^6 - 12z^3y^4 + 16y^8)\).

Key Concepts

Sum of Cubes Formula

The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). It is used to factor expressions where two terms are perfect cubes added together. Recognizing the terms as cubes allows you to apply this formula to factor the polynomial completely.

Identifying Perfect Powers

To factor using special formulas, you must identify if terms are perfect powers, such as perfect squares or cubes. For example, 27z⁹ is (3z³)³ and 64y¹² is (4y⁴)³. Recognizing these helps in rewriting the polynomial in a form suitable for applying the sum of cubes formula.

Polynomial Factoring Techniques

Factoring polynomials involves breaking them down into simpler polynomials multiplied together. Techniques include factoring out the greatest common factor, using special product formulas, and recognizing patterns. Mastery of these techniques is essential for simplifying expressions and solving polynomial equations.