Problem 42 - Elementary Algebra

Question

Factor completely. $$ 9 x 2+5 x y-12 y 2 $$

Step-by-Step Solution

Answer

The expression \(9x^2 + 5xy - 12y^2\) factors to \(3(x - y)(3x + 4y)\).

Step 1: Rearrange the Terms

The expression given is \( 9x^2 + 5xy - 12y^2 \). To make it easier to factor, let's rewrite the expression so that it clearly reflects the standard form \( ax^2 + bxy + cy^2 \). We already have this form: \( 9x^2 + 5xy - 12y^2 \). In this case, no rearrangement is necessary as the terms are already organized correctly.

Step 2: Identify the Coefficients

Take note of the coefficients in the quadratic trinomial: \( a = 9 \), \( b = 5 \), and \( c = -12 \). These will help us factor the trinomial.

Step 3: Multiply \( a \times c \)

Multiply the leading coefficient \( a = 9 \) by the constant term \( c = -12 \) to find the product: \( 9 \times -12 = -108 \).

Step 4: Find Factors of \(-108\) that Add to \(b\)

Look for two numbers that multiply to \(-108\) and add to \(b = 5\). These numbers are \(12\) and \(-9\) because \(12 imes (-9) = -108\) and \(12 + (-9) = 3\). Wait, instead, we should find two numbers multiplying to \(-108\) and adding to \(5\), which are actually \(12\) and \(-9\).

Step 5: Rewrite the Middle Term

Use the numbers 12 and -9 to rewrite the middle term \(5xy\) as two separate terms: \(12xy - 9xy\). Therefore, the expression becomes \(9x^2 + 12xy - 9xy - 12y^2\).

Step 6: Group Terms

Group the expression into two pairs: \((9x^2 + 12xy) + (-9xy - 12y^2)\).

Step 7: Factor Each Group Separately

Factor out the greatest common factor from each pair. For the first group \(9x^2 + 12xy\), factor out \(3x\) to get \(3x(3x + 4y)\). For the second group \(-9xy - 12y^2\), factor out \(-3y\) to get \(-3y(3x + 4y)\).

Step 8: Factor Out the Common Binomial

Notice that both terms include \((3x + 4y)\). Factor this common binomial out from the two groups to get \((3x - 3y)(3x + 4y)\).

Step 9: Simplify Further if Possible

Notice that in \((3x - 3y)(3x + 4y)\), the first factor can be further factored by taking out 3: \(3(x - y)(3x + 4y)\). This is the fully factored form.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a special type of polynomial that consists of three terms, specifically involving variables raised to the second degree. The general form of a quadratic trinomial is usually written as \(ax^2 + bxy + cy^2\), where:

• \(a\), \(b\), and \(c\) are coefficients
• \(x\) and \(y\) are variables
• \(ax^2\) is the quadratic term because of the squared variable

In our exercise, the trinomial is \(9x^2 + 5xy - 12y^2\). This expression is already in the right form, with each term clearly identified. Understanding that these terms represent parts of a quadratic trinomial helps in identifying the coefficients and using them to factor the expression. Quadratic trinomials can often be simplified or factored into the product of two binomials, making calculations easier and solving equations straightforward.

Greatest Common Factor

The Greatest Common Factor (GCF) is crucial when dealing with polynomials, especially when attempting to factor them. In simple terms, the GCF is the largest number or expression that divides each term in a polynomial without leaving a remainder. Identifying the GCF is a critical step in simplifying expressions or factoring them completely. To find the GCF, you should:

• Examine each term in the expression
• Look for numerical coefficients and variable factors that appear in each term
• Choose the highest powers of each variable common in the terms

For example, in the expression \(9x^2 + 12xy\), the GCF is \(3x\) because 3 is the highest number that can divide both terms, and \(x\) is found in both terms. Similarly, in \(-9xy - 12y^2\), the GCF is \(-3y\). Factoring out the GCF helps in breaking down complex polynomials and revealing simpler expressions or binomials such as in this problem.

Standard Form of Quadratics

The standard form of a quadratic expression lays the foundation for solving and factoring these types of polynomials effectively. This form is typically written as \(ax^2 + bxy + cy^2\) for expressions involving two variables, just like in our given problem: \(9x^2 + 5xy - 12y^2\).The importance of writing an expression in its standard form includes:

• Bringing clarity to the structure of the polynomial
• Identifying the coefficients necessary for further calculations
• Helping in recognizing patterns for factoring

By starting with this form, the process of factoring becomes systematic. The coefficients \(a\), \(b\), and \(c\) are used to guide the steps necessary for finding two numbers that fulfill the conditions for factoring, such as multiplying to a certain product or adding to a specific sum. In this exercise, recognizing the coefficients allowed us to efficiently deconstruct the trinomial into its factored form.