Problem 106 - Algebra 1: Concepts and Skills

Question

Factor the trinomial. (Lessons 10.5,10.6) $$ 14 x^{2}-19 x-3 $$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(14x^{2} -19x -3\) is \((2x - 3)(7x + 1)\).

Step 1: Identify the coefficients and the constant

The trinomial given is \(14x^{2} - 19x - 3\). The coefficient of \(x^{2}\) is \(a = 14\), the coefficient of \(x\) is \(b = -19\), and the constant term is \(c = -3\).

Step 2: Find two numbers

Look for two numbers such that their product is equal to the product of the coefficient of \(x^{2}\) and the constant term (in this case, \(14*(-3) = -42\)) and their sum is equal to the coefficient of \(x\) (in this case, -19). The numbers are -21 and 2, since \(-21 * 2 = -42\) and \(-21 + 2 = -19\).

Step 3: Rewrite the equation

Rewrite the trinomial by breaking down the middle term using the two numbers found in step 2. This results in: \(14x^{2} -21x +2x - 3\).

Step 4: Factor by grouping

Group the terms two by two, and factor out the greatest common factor in each group. This gives: \(7x(2x - 3) + 1(2x - 3)\).

Step 5: Write the final factorized form

Both groups have a common factor of \(2x - 3\). Factoring this out gives the final answer: \((2x - 3)(7x + 1)\).

Key Concepts

Coefficient Identification

Understanding how to identify coefficients in a trinomial is a fundamental skill in algebra. Coefficients are the numerical parts of the terms in an algebraic expression. In the given trinomial, 14x^{2} - 19x - 3, we must pinpoint each term's coefficient.

Here, a = 14 is the coefficient of the x^{2} term, signifying it's a quadratic trinomial. The coefficient b = -19 accompanies the linear x term, and the constant c = -3 stands alone without a variable. Identification of these coefficients is crucial as they play a key role in the subsequent steps of factoring.

Factor by Grouping

Factor by grouping is a technique used when direct factoring is not possible. It requires the manipulation of the middle term of a trinomial, breaking it into two terms that allow creating groups with a common factor.

In our exercise, after discovering the numbers -21 and 2, we rewrite the trinomial as 14x^{2} -21x + 2x - 3. We then separate the terms into two groups: (14x^{2} -21x) and (2x - 3). From here, we find the greatest common factor in each pair: 7x from the first and 1 (or simply nothing) from the second. Factoring these out results in two identical binomials (2x - 3), signifying the grouping method's success and enabling us to factor the trinomial efficiently.

Finding Common Factors

After completing the grouping process, we often encounter a scenario where both groups share a common factor. In such cases, this shared factor can be factored out, simplifying the expression further.

In our grouped equation, 7x(2x - 3) + 1(2x - 3), the binomial (2x - 3) appears in both terms. This binomial is the common factor. By factoring (2x - 3) out, we are left with (2x - 3)(7x + 1), which represents the original trinomial's factors. Recognizing and factoring out the common factor is a crucial final step in solving the factoring problem and cannot be overlooked.