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Magazine Chapter 3: Problem 33
For Problems \(31-56\), factor completely each of the trinomials and indicate any that are not factorable using integers. $$ 12 x^{2}-x-6 $$
Short Answer
Expert verified
The trinomial \(12x^2 - x - 6\) factors as \((3x + 2)(4x - 3)\).
Step by step solution
01
Identify the trinomial
The trinomial given is \(12x^2 - x - 6\). Our task is to factor it completely.
02
Find factors of the product of the first and last coefficients
Multiply the first coefficient (12) and the last coefficient (-6) to get the product: \(12 \times (-6) = -72\). We need to find two numbers that multiply to -72 and add to -1 (the middle coefficient).
03
Determine the factor pair
The factor pair of -72 that adds to -1 is 8 and -9, because \(8 \times (-9) = -72\) and \(8 + (-9) = -1\).
04
Rewrite the middle term
Rewrite \(-x\) using the two numbers: \(12x^2 + 8x - 9x - 6\).
05
Factor by grouping
Group the terms: \((12x^2 + 8x) + (-9x - 6)\).
06
Factor each group
Factor out the greatest common factor from each group: \(4x(3x + 2) - 3(3x + 2)\).
07
Factor out the common binomial factor
Notice that \((3x + 2)\) is common in both terms, so factor it out: \((3x + 2)(4x - 3)\).
08
Verify the factorization
Expand \((3x + 2)(4x - 3)\) to check your work: \(3x imes 4x = 12x^2\), \(3x imes -3 = -9x\), \(2 imes 4x = 8x\), \(2 imes -3 = -6\). Combine the middle terms to get \(-9x + 8x = -x\), confirming the factorization is correct.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factor by Grouping
Factor by grouping is a method used to simplify an expression or to find its factors by organizing terms into groups with common factors. In the problem with the trinomial \(12x^2 - x - 6\), we aim to rewrite it in a way that makes factoring simpler.
- We first look to split the middle term into two numbers that multiply to the product of the leading and constant terms, and add to the middle coefficient.
- Once split, terms are grouped into sets that can individually be factored more easily.
Quadratic Expression
A quadratic expression is a polynomial of the form \(ax^2 + bx + c\). These expressions are characterized by the highest power of the variable \(x\) being 2. In our case, the given quadratic expression is \(12x^2 - x - 6\).
- The expression has three terms: a quadratic term \(12x^2\), a linear term \(-x\), and a constant term \(-6\).
- The objective when working with such expressions can include simplifying, factoring, or solving them in the context of quadratic equations.
Factoring Trinomials
Factoring trinomials involves breaking down a quadratic expression into a product of two binomials. The task is to identify two numbers stemming from specific conditions: they must multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.
- Consider the trinomial \(12x^2 - x - 6\). Multiply the leading coefficient (12) by the constant (-6), giving us -72.
- Find two numbers that multiply to -72 and add up to -1, which are 8 and -9 in this case.
Polynomial Factorization
Polynomial factorization is the process of decomposing a polynomial into a product of simpler, more manageable factors. This process can simplify solving equations and understanding polynomial properties.
- Start by recognizing the form of expression you are working with, like the trinomial in this exercise.
- Employ the method of factoring by grouping for polynomials with more than three terms. For trinomials, use approaches like finding factor pairs.
