Problem 69 - Algebra 1

Question

Factor the expression. $$-48 x^{2}+216 x-243$$

Step-by-Step Solution

Answer

The factor of the expression \(-48x^{2}+216x-243\) is \(-3(4x-9)^2\)

Step 1: Identify and Factor out the Greatest Common Factor (GCF)

The GCF among the three terms \(-48x^{2}\), \(216x\), and \(-243\) is \(-3\). To factor it out, divide each term by \(-3\), which yields \[16x^2-72x+81\]. Thus, the initial expression becomes \(-3(16x^2-72x+81)\].

Step 2: Factor the quadratic part

The quadratic \(16x^2-72x+81\) can be factored into \((4x-9)^2\), a perfect square. To see this, recognize that 16, the coefficient of \(x^2\), is \(4^2\), and 81 is \(9^2\), and \(-2*4*9\) equals \(-72\), the coefficient of x. Hence, we can rewrite the quadratic as \((4x-9)^2\).

Step 3: Combine the factors

Combine the factored GCF from Step 1 with the factored quadratic from Step 2. The fully factored expression is: \[-3(4x-9)^2\]

Key Concepts

Greatest Common Factor (GCF)

Factoring out the Greatest Common Factor (GCF) is often the first step in simplifying and factoring more complex polynomial expressions. The GCF is the largest factor that divides all the terms in a polynomial evenly. For example, in the polynomial

• \(-48x^2 + 216x - 243\), the greatest common factor that can be divided into each term is \(-3\).

Once you identify the GCF, you can "factor it out" by dividing every term in the polynomial by this factor. In our example, dividing each term by \(-3\) we get \(16x^2 - 72x + 81\).

Identifying the GCF helps simplify calculations in subsequent steps of factoring and can lead to spotting other patterns, like perfect square trinomials.

Perfect square trinomials

A perfect square trinomial is a specific type of trinomial expression which can be expressed as the square of a binomial. In general terms, it takes the form \((ax + b)^2\), which expands to \(a^2x^2 + 2abx + b^2\).

Identifying a perfect square trinomial requires checking if the first and last terms are perfect squares and if the middle term fits the double product rule. In the expression \(16x^2 - 72x + 81\), we notice:

• \(16x^2\) is \((4x)^2\)
• \(81\) is \(9^2\)
• The middle term \(-72x\) is twice the product of \(4x\) and \(9\)

After verification, it's clear that \((4x - 9)^2\) is the factored form. Recognizing this pattern accelerates the process and ensures more efficient factoring of quadratic expressions.

Quadratic expressions

Quadratic expressions are polynomial expressions of degree two, typically written in the form \(ax^2 + bx + c\). Factoring these expressions often involves techniques such as using the GCF, completing the square, or recognizing special products like perfect square trinomials.

Understanding the structure of quadratic expressions is fundamental. For instance, the quadratic \(16x^2 - 72x + 81\) was simplified by first identifying the GCF and then recognizing it as a perfect square trinomial. This approach highlights how multiple factoring techniques can be nested together to simplify even complicated expressions.

Ultimately, mastering how to manipulate quadratic expressions through factoring is crucial for solving quadratic equations and for more advanced topics in algebra.