Factor each polynomial. See Example 7. (x-4)3+64

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 88

1장, 문제 88

Factor each polynomial. See Example 7. (x-4)³+64

검증된 단계별 안내

Recognize that the expression $ (x-4)^3 + 64 $ is a sum of cubes, since $64$ can be written as \$4^3$. So the expression is of the form $a^3 + b^3$ where $a = (x-4)$ and $b = 4$.

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

Apply the formula by substituting $a = (x-4)$ and $b = 4$ into the factors: first factor is $(x-4 + 4)$ and the second factor is $( (x-4)^2 - (x-4)(4) + 4^2 )$.

Simplify the first factor: $(x-4 + 4)$ simplifies to $x$. Then expand and simplify the second factor: expand $(x-4)^2$, multiply $(x-4)(4)$, and calculate \$4^2$.

Write the fully factored form as the product of the simplified first factor and the simplified second factor.

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 each perfect cubes added together. Recognizing (x - 4)³ + 64 as a sum of cubes allows us to apply this formula directly.

Identifying Perfect Cubes

A perfect cube is a number or expression raised to the third power, such as x³ or 64 (since 64 = 4³). Identifying each term as a perfect cube is essential before applying the sum or difference of cubes formulas in polynomial factoring.

Polynomial Factoring Techniques

Factoring polynomials involves rewriting them as products of simpler polynomials. Techniques include recognizing special patterns like sum/difference of cubes, difference of squares, and factoring by grouping. Mastery of these methods simplifies solving polynomial equations.