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

Verified step by step guidance

1. Identify the trinomial to factor: \$5a^2 - 7ab - 6b^2$.
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 \(-7\).
3. Find the pair of numbers that satisfy these conditions. In this case, the numbers are \(-10\) and \(3\) because \(-10 \times 3 = -30\) and \(-10 + 3 = -7\).
4. Rewrite the middle term \(-7ab\) as \(-10ab + 3ab\) to split the trinomial: \$5a^2 - 10ab + 3ab - 6b^2$.
5. 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)$.

Key Concepts

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.

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.

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.