Problem 65 - Algebra 2

Question

Factor completely. $$ 6 x^{2}-14 x-12 $$

Step-by-Step Solution

Answer

The completely factored form of the polynomial is \( 2(x - 3)(3x + 2) \).

Step 1: Identify Common Factors

Start by identifying the greatest common factor (GCF) of all the coefficients in the polynomial. Here, the coefficients are 6, -14, and -12. The greatest common factor of these numbers is 2. Factor out the 2 from the polynomial.After factoring out 2, the expression becomes: \[ 2(3x^2 - 7x - 6) \]

Step 2: Factor the Quadratic Expression

We need to factor the quadratic expression \(3x^2 - 7x - 6\) inside the parentheses. To do this, look for two numbers that multiply to \(3 \times -6 = -18\) and add up to \(-7\). These numbers are 2 and -9.

Step 3: Rewrite the Middle Term

Rewrite the quadratic expression by splitting the middle term, \(-7x\), using the numbers found in the previous step: 2 and -9. So, the expression becomes:\[3x^2 + 2x - 9x - 6\]

Step 4: Group and Factor by Grouping

Group the first two terms and the last two terms separately:\[ (3x^2 + 2x) + (-9x - 6) \]Next, factor out the greatest common factor from each group:- From \(3x^2 + 2x\), factor out \(x\): \(x(3x + 2)\)- From \(-9x - 6\), factor out \(-3\): \(-3(3x + 2)\)

Step 5: Factor Out the Common Binomial

Both groups have a common binomial factor, \((3x + 2)\). Factor this out to get:\[ x(3x + 2) - 3(3x + 2) = (x - 3)(3x + 2) \]

Step 6: Write the Complete Factored Form

Combine the factored common factor from Step 1 with the result of Step 5 to write the fully factored form of the original polynomial:\[ 2(x - 3)(3x + 2) \]

Key Concepts

Greatest Common Factor

When working with polynomials, you often have to start by identifying the greatest common factor (GCF) of the terms. The GCF is the largest number that divides all the coefficients of the given polynomial terms evenly. In our polynomial, the terms have coefficients of 6, -14, and -12. The greatest common factor among these is 2.
Factoring out the GCF simplifies the expression, making it easier to work with. By dividing each term by the GCF, you pull this factor out in front of the parentheses. For example, factoring out a 2 from the polynomial expression leads us to:

• \[ 2(3x^2 - 7x - 6) \]

This step might seem small, but it significantly reduces complexity and sets the stage for effective factoring later on.

Quadratic Expression

A quadratic expression is a polynomial where the highest exponent of the variable (usually denoted as \(x\)) is 2. In our exercise, the quadratic expression is found once the greatest common factor is pulled out, leaving us with \(3x^2 - 7x - 6\).
Quadratic expressions are in the standard form:

• \( ax^2 + bx + c \)

Here, \( a, b, \) and \( c \) represent coefficients, with \( a \) not equal to zero. In the expression \(3x^2 - 7x - 6\), \(a = 3\), \(b = -7\), and \(c = -6\).
Factoring a quadratic requires finding two numbers that multiply to \( a \times c \) and add up to \( b \). This process helps to break down the expression into simpler binomial factors.

Factoring by Grouping

Factoring by grouping is a useful technique when dealing with polynomials, particularly those that aren't straightforward to factor using simple methods. Once you have split the middle term using numbers that satisfy the multiplication and addition condition, the expression is easier to handle. In our case:

• \[ 3x^2 - 7x - 6 \rightarrow 3x^2 + 2x - 9x - 6 \]

The expression is grouped into pairs:

• \( (3x^2 + 2x) + (-9x - 6) \)

Each group should have a common factor that can be factored out. From the first group, \( x \) is factored out, and from the second, \(-3\). This leads to:

• \( x(3x + 2) - 3(3x + 2) \)

When both groups include the common binomial, \(3x + 2\), it can be factored out, yielding:

• \( (x - 3)(3x + 2) \)

This process, which might seem intricate at first, offers a strategic approach to simplifying complex quadratic expressions.