In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x²−11x+4

검증된 단계별 안내

Identify the trinomial: \(6x^2 - 11x + 4\). The goal is to factor it into the form \((ax + b)(cx + d)\), where \(a\), \(b\), \(c\), and \(d\) are constants.

Multiply the leading coefficient (6) and the constant term (4): \(6 \times 4 = 24\). Now, find two numbers that multiply to 24 and add to the middle coefficient, \(-11\). These numbers are \(-8\) and \(-3\).

Rewrite the middle term \(-11x\) as \(-8x - 3x\): \(6x^2 - 8x - 3x + 4\). This step splits the trinomial into four terms to allow factoring by grouping.

Group the terms into two pairs: \((6x^2 - 8x) - (3x - 4)\). Factor out the greatest common factor (GCF) from each group: \(2x(3x - 4) - 1(3x - 4)\).

Notice that \((3x - 4)\) is a common factor. Factor it out: \((2x - 1)(3x - 4)\). This is the factored form of the trinomial.

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 Trinomials

Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is crucial for simplifying expressions and solving equations.

Prime Trinomials

A prime trinomial is a quadratic expression that cannot be factored into simpler binomial expressions with rational coefficients. Recognizing when a trinomial is prime is essential, as it indicates that the expression cannot be simplified further. This concept helps in determining the nature of the roots of the quadratic equation.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. This formula is particularly useful when factoring is difficult or when determining if a trinomial is prime. Understanding how to apply the quadratic formula can help verify the results of factoring.

In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x2−11x+4