Problem 31 - Algebra 2

Question

Factor each expression. $$ 3 x^{2}+7 x-20 $$

Step-by-Step Solution

Answer

(3x - 5)(x + 4)

Step 1: Identify Coefficients

First, identify the coefficients of the quadratic trinomial. In this case, the coefficient of the quadratic term, 'a', is 3, the coefficient of the linear term, 'b', is 7, and the constant term, 'c', is -20.

Step 2: Determine the Product AC

Multiply the coefficients 'a' and 'c' to determine the product AC. Here, AC = 3 * (-20) = -60.

Step 3: Find Two Numbers

The goal is to find two numbers that multiply to AC (-60) and add up to 'b' (+7). These two numbers are 12 and -5 since 12 * (-5) = -60 and 12 + (-5) = 7.

Step 4: Rewrite the Linear Term

Rewrite the linear term of the quadratic trinomial using the two numbers found in the previous step, splitting the 7x into two terms: 12x and -5x.

Step 5: Group the Terms

Group the terms into two pairs: (3x^2 + 12x) and (-5x - 20).

Step 6: Factor by Grouping

Factor each pair from the previous step. For the first group, factor out the greatest common factor, which is 3x, to get 3x(x + 4). In the second group, factor out the greatest common factor, which is -5, to get -5(x + 4).

Step 7: Factor Out the Common Binomial

The common binomial factor from both groups is (x + 4). Factor this out to get the final factored form of the expression: (3x - 5)(x + 4).

Key Concepts

Understanding Quadratic Trinomials

When working with quadratic trinomials, you're dealing with a type of polynomial that has a degree of two. These expressions are in the form of ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. For the quadratic trinomial 3x^2 + 7x - 20, we notice that it has three terms - a squared term, a linear term, and a constant.

The significance of factoring such an expression is rooted in finding its roots, the values of x for which the expression equals to zero. It's also a crucial step in graphing the parabola represented by the equation, simplifying algebraic fractions, or solving quadratic equations.

For many students, the concept may initially seem daunting. However, breaking it down into smaller, understandable steps makes the process manageable. Through practice, factoring quadratic trinomials becomes an invaluable tool in solving a variety of mathematical problems.

Coefficient Identification

Coefficient identification is fundamental to understanding and manipulating algebraic expressions, especially quadratics. In our expression 3x^2 + 7x - 20, the coefficients are numerical values attached to the terms of the polynomial. Specifically, the coefficient of the quadratic term a is 3, while the coefficient of the linear term b is 7. Lastly, the constant term c stands alone as -20.

Recognizing and working with these coefficients is a vital skill because they not only dictate the shape and position of the graph of the quadratic function but also play a central role in methods used to manipulate and factor these expressions. The relationship between these coefficients can reveal much about the nature of the quadratic, such as its concavity and the axis of symmetry.

The AC Method

The AC method is an efficient process used to factor quadratic trinomials, and it hinges upon the concept of coefficient identification. Once you've determined the coefficients, which for our example are a = 3, b = 7, and c = -20, you'll multiply a and c to find the product AC. This step is crucial as it prepares the ground for finding two special numbers.

These two numbers must multiply together to give you AC, and at the same time, add up to the linear coefficient b. It's like solving a little puzzle - the product must equal AC (in our case, -60), and their sum equals b (which is 7). Once you find these two numbers, you will use them to decompose the middle term and proceed to factor by grouping.

In our example, 12 and -5 satisfy both conditions as their product is -60 and their sum is 7. This decomposes the middle term and leads us through the next steps of grouping and factoring which ultimately simplifies the initially daunting-looking quadratic trinomial into an approachable and solvable form.