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

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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