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Ch. R - Review of Basic Concepts
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. R - Review of Basic ConceptsProblem 45
Chapter 1, Problem 45

Factor each trinomial, if possible. See Examples 3 and 4. 5a2-7ab-6b2

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1
Identify the trinomial to factor: \(5a^2 - 7ab - 6b^2\).
Look for two numbers that multiply to the product of the first and last coefficients (5 and -6), which is \(5 \times (-6) = -30\), and add up to the middle coefficient, which is \(-7\).
Find the pair of numbers that satisfy these conditions. In this case, the numbers are \(-10\) and \(3\) because \(-10 \times 3 = -30\) and \(-10 + 3 = -7\).
Rewrite the middle term \(-7ab\) as \(-10ab + 3ab\) to split the trinomial: \(5a^2 - 10ab + 3ab - 6b^2\).
Group the terms in pairs and factor each group: \((5a^2 - 10ab) + (3ab - 6b^2)\), then factor out the greatest common factor from each group to get \(5a(a - 2b) + 3b(a - 2b)\), and finally factor out the common binomial factor \((a - 2b)\) to write the expression as \((a - 2b)(5a + 3b)\).

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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 of the form ax^2 + bx + c as a product of two binomials. This process helps simplify expressions and solve equations. Recognizing patterns and using methods like trial and error or the AC method are common approaches.
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The AC Method

The AC method is a technique for factoring trinomials when the leading coefficient (a) is not 1. It involves multiplying a and c, finding two numbers that multiply to ac and add to b, then splitting the middle term accordingly to factor by grouping.
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Factoring by Grouping

Factoring by grouping is a method used after splitting the middle term in a trinomial. It involves grouping terms in pairs, factoring out the greatest common factor from each group, and then factoring out the common binomial factor to express the polynomial as a product of binomials.
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