Factor each polynomial. See Examples 5 and 6. $$27z9+64y1227z^9$+64y^{12}$$

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 78

1장, 문제 78

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

검증된 단계별 안내

Recognize that the polynomial $27z^9 + 64y^{12}$ is a sum of two cubes because $27z^9 = (3z^3)^3$ and $64y^{12} = (4y^4)^3$.

Recall the sum of cubes factoring formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Identify $a = 3z^3$ and $b = 4y^4$ in the expression $27z^9 + 64y^{12}$.

Apply the formula by substituting $a$ and $b$: write the factorization as $(3z^3 + 4y^4)((3z^3)^2 - (3z^3)(4y^4) + (4y^4)^2)$.

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 $(3z^3 + 4y^4)(9z^6 - 12z^3y^4 + 16y^8)$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Factor each polynomial. See Examples 5 and 6. 27z9+64y1227z^9+64y^{12}