Problem 64 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(12 x^{3}-34 x^{2}+24 x\)

Step-by-Step Solution

Answer

\(x \cdot 2(3x - 4)(2x - 3)\)

Step 1: Identify Common Factors

First, look at all the terms in the polynomial to find any common factors. In the trinomial \(12x^3 - 34x^2 + 24x\), each term contains the factor \(x\). Therefore, the greatest common factor (GCF) of the terms is \(x\).

Step 2: Factor Out the GCF

Factor \(x\) out of each term in the polynomial, giving us \(x(12x^2 - 34x + 24)\). Now we need to factor the quadratic expression \(12x^2 - 34x + 24\) inside the parentheses.

Step 3: Apply the AC Method

Multiply the coefficients \(a\) and \(c\) from the quadratic \(12x^2 - 34x + 24\), where \(a = 12\) and \(c = 24\). This gives \(12 \times 24 = 288\). We need two numbers that multiply to 288 and add to \(-34\).

Step 4: Find Suitable Pair of Numbers

After testing different factorizations of 288, we find that \(-18\) and \(-16\) multiply to 288 and add to \(-34\).

Step 5: Rewrite the Middle Term

Use the pair of numbers \(-18\) and \(-16\) to rewrite the middle term \(-34x\). This gives us \(12x^2 - 18x - 16x + 24\).

Step 6: Factor by Grouping

Group the terms into two pairs: \((12x^2 - 18x)\) and \((-16x + 24)\). Factor out the greatest common factor from each pair: \(6x(2x - 3) - 8(2x - 3)\). Notice that \(2x - 3\) is a common factor.

Step 7: Factor Out the Common Binomial

Since \(2x - 3\) is common in both terms, factor it out: \((6x - 8)(2x - 3)\).

Step 8: Simplify the Expression

Now factor \(6x - 8\) completely. Notice that \(6x - 8\) has a common factor of 2. Factor it out to obtain \(2(3x - 4)\). Thus, we have the complete factorization: \(x \times 2(3x - 4)(2x - 3)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor shared by all terms in a polynomial. Finding the GCF is an essential first step in factoring polynomials because it simplifies the expression. By identifying and factoring out the GCF, we reduce the polynomial to an easier form, helping us focus on more complex factorization techniques later.

To find the GCF, look at the coefficients and variables in each term. Check for the highest number and variable power present in every term. In the expression \(12x^3 - 34x^2 + 24x\):

• Coefficients: The coefficients 12, 34, and 24 have a common factor of 2.
• Variables: Each term contains at least one \(x\). Therefore, \(x\) is a common factor.

By factoring out \(2x\), we simplify the trinomial, making subsequent steps more straightforward.

Quadratic Expressions

A quadratic expression is any expression of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratics form the backbone of many algebraic applications, making understanding them crucial.

In our exercise, once we factor out the GCF, we're left with the quadratic expression \(12x^2 - 34x + 24\). Here:

• \(a = 12\), the coefficient of \(x^2\).
• \(b = -34\), the coefficient of \(x\).
• \(c = 24\), the constant term.

Understanding the structure of quadratic expressions allows us to apply various methods, like the AC method, for further factorization.

AC Method

The AC method is a systematic approach to factor quadratic expressions, particularly when the simple guess-and-check methods are challenging. It involves multiplying the first coefficient (\(a\)) by the constant term (\(c\)) to help find two numbers that can be used to break down the middle term, thus making factorization more manageable.

In our example, \(a = 12\) and \(c = 24\), so we calculate the product \(12 \times 24 = 288\). We must identify two numbers that multiply to 288 and add to the middle term's coefficient, \(-34\). After trying different combinations, we find \(-18\) and \(-16\) satisfy both conditions. This pair of numbers helps to decompose the middle term, enabling us to move to the next step: factoring by grouping.

Factoring by Grouping

Factoring by grouping is a technique used to factor polynomial expressions by rearranging and grouping terms. This method is particularly useful when dealing with four terms because it allows us to take advantage of common factors and simplify the polynomial into binomial factors.

In our exercise, after using the AC method, we rewrote \(-34x\) as \(-18x - 16x\), giving us four terms: \(12x^2 - 18x - 16x + 24\). We group these into two pairs:

• First pair: \((12x^2 - 18x)\)
• Second pair: \((-16x + 24)\)

Each pair can be factored individually by finding their respective common factors:

• From the first pair, factor out \(6x\), resulting in \(6x(2x - 3)\).
• From the second pair, factor out \(-8\), resulting in \(-8(2x - 3)\).

Since \(2x - 3\) is common to both, we factor it out, resulting in a clean, simplified expression \((6x - 8)(2x - 3)\), which can be further simplified by factoring out any additional common factors.