Problem 62 - Elementary Algebra

Question

Factor out a negative common factor first and then factor further if possible. $$ -18 x 3-51 x 2+9 x $$

Step-by-Step Solution

Answer

Factorized expression: \(-3x(6x-1)(x+3)\).

Step 1: Identify the Negative Common Factor

First, look at the coefficients of the expression \(-18x^3 - 51x^2 + 9x\). Notice that each coefficient is divisible by 3, and the expression starts with a negative term. Therefore, the greatest common factor (GCF) is \(-3x\).

Step 2: Factor Out the Negative Common Factor

Factor out \(-3x\) from each term of the expression: \[-18x^3 - 51x^2 + 9x = -3x(6x^2 + 17x - 3)\]

Step 3: Factor Inside the Parenthesis

Now, focus on factoring the quadratic polynomial \(6x^2 + 17x - 3\). We want two numbers that multiply to \(ac = 6(-3) = -18\)\ and add to the middle coefficient \(b = 17\).

Step 4: Decompose and Pair the Middle Term

The numbers \(-1\)\ and \(18\)\ multiply to \(-18\)\ and add to \(17\)\. Rewrite \(6x^2 + 17x - 3\) as:\[6x^2 - x + 18x - 3\]

Step 5: Factor by Grouping

Group terms to factor separately:\[(6x^2 - x) + (18x - 3)\]Factor out the GCF in each group:\[x(6x - 1) + 3(6x - 1)\]

Step 6: Factor Out the Common Binomial Factor

Since \((6x - 1)\) is common in both groups, factor it out:\[(6x - 1)(x + 3)\]

Step 7: Put the Factorization Together

Combine the results from factoring out the negative common factor and the quadratic:\[-3x(6x^2 + 17x - 3) = -3x(6x - 1)(x + 3)\]

Key Concepts

Greatest Common Factor

Finding the greatest common factor (GCF) is a handy tool when working with polynomials. It helps to simplify expressions by identifying the largest factor shared by all the terms. In the polynomial expression \(-18x^3 - 51x^2 + 9x\), you'll notice that each coefficient \(-18, -51,\) and \(9\) shares a common factor of \(3\). Since the expression starts with a negative term, it's important to consider the negative sign as part of the GCF. Hence, the GCF here is \(-3x\).

• Step 1: Identify shared factors.
• Step 2: Consider the sign of the leading term.

Factoring out the GCF simplifies the problem into smaller, more manageable parts.

Quadratic Factoring

Quadratic factoring can seem challenging, but with practice, it becomes more intuitive. Once the expression is simplified by factoring out the GCF, you'll often encounter a quadratic expression, like \(6x^2 + 17x - 3\). To factor this, we need to find two numbers that multiply to \(-18\) (the product of \(a\) and \(c\), where \(a = 6\) and \(c = -3\)) and add to \(b = 17\).For this example, the numbers are \(-1\) and \(18\). This step involves:

• Finding numbers that fit the criteria.
• Rewriting the middle term using these numbers.

Mastering this not only helps with factoring quadratics but strengthens your overall understanding of polynomials.

Polynomial Expression

A polynomial expression is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient. In our example, \(-18x^3 - 51x^2 + 9x\), you see three terms with descending powers of \(x\).

• The expression is made more manageable by identifying and factoring out common factors.
• Breaking down polynomials step-by-step helps in understanding the structure.

Recognizing the structure allows you to approach factoring with confidence and clarity.

Factoring by Grouping

Factoring by grouping is a useful technique when handling polynomials with four or more terms. The key is to group terms together so that they share a common factor. In our expression \(6x^2 - x + 18x - 3\), we can group and factor as follows:

• Group terms: \((6x^2 - x) + (18x - 3)\).
• Factor each group: \(x(6x - 1) + 3(6x - 1)\).

Notice that \((6x - 1)\) is common in both groups. Factoring by grouping often involves rearranging terms strategically, making it a flexible and powerful technique in polynomial division.