Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. $5 x^{4} - 20 x^{2} = 0$

검증된 단계별 안내

Start by writing the given polynomial equation: $5x^4 - 20x^2 = 0$.

Factor out the greatest common factor (GCF) from both terms. Identify the GCF of \$5x^4$ and $20x^2$, which is $5x^2$. So, factor it out: $5x^2(x^2 - 4) = 0$.

Recognize that the expression inside the parentheses, $x^2 - 4$, is a difference of squares. Factor it further as $(x - 2)(x + 2)$, so the equation becomes $5x^2(x - 2)(x + 2) = 0$.

Apply the zero-product principle, which states that if a product of factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero: $5x^2 = 0$, $x - 2 = 0$, and $x + 2 = 0$.

Solve each equation separately: For $5x^2 = 0$, divide both sides by 5 and solve for $x$. For $x - 2 = 0$, add 2 to both sides. For $x + 2 = 0$, subtract 2 from both sides. These solutions give the roots of the original polynomial.

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.

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or expressions. For the given equation, common factors like powers of x and constants can be factored out to simplify the equation, making it easier to solve.

Zero-Product Principle

The zero-product principle states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for the variable.

Solving Polynomial Equations

Solving polynomial equations involves finding all values of the variable that satisfy the equation. After factoring and applying the zero-product principle, each resulting equation is solved individually to find all possible roots.

Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle. 5x4−20x2=05x^4 - 20x^2 = 0