Problem 51 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(8 x^{2}+6 x y-27 y^{2}\)

Step-by-Step Solution

Answer

The trinomial \(8x^2 + 6xy - 27y^2\) factors as \((2x - 3y)(4x + 9y)\).

Step 1: Identify the form of the trinomial

The given trinomial is in the form of a quadratic expression: \(ax^2 + bxy + cy^2\), where \(a = 8\), \(b = 6\), and \(c = -27\). This suggests that we will be looking for two binomials that multiply together to give the original trinomial.

Step 2: Check for common factors

Examine if there is any greatest common factor (GCF) for all the coefficients \(8\), \(6\), and \(-27\). In this case, there are no common factors, so no factoring out is needed.

Step 3: Factor by grouping

To factor by grouping, we need to split the middle term into two terms such that their product equals \(ac = 8 \times (-27) = -216\) and their sum equals \(b = 6\). We need two numbers that multiply to \(-216\) and add to \(6\). These numbers are \(18\) and \(-12\). Rewrite the trinomial as: \(8x^2 + 18xy - 12xy - 27y^2\).

Step 4: Group the terms

Group the four terms into two pairs: \((8x^2 + 18xy) + (-12xy - 27y^2)\).

Step 5: Factor out the greatest common factor from each pair

Factor the first group:\[8x^2 + 18xy = 2x(4x + 9y)\]. Factor the second group:\[-12xy - 27y^2 = -3y(4x + 9y)\].

Step 6: Factor out the common binomial

Notice that each group has a common binomial factor, \(4x + 9y\). Therefore, we can factor this out:\[(2x - 3y)(4x + 9y)\].

Step 7: Verify the factorization

Multiply the factors \((2x - 3y)(4x + 9y)\) to ensure they result in the original trinomial \(8x^2 + 6xy - 27y^2\). Use the distributive property (FOIL) to check the multiplication: \(2x \cdot 4x + 2x \cdot 9y - 3y \cdot 4x - 3y \cdot 9y = 8x^2 + 18xy - 12xy - 27y^2 = 8x^2 + 6xy - 27y^2\). This confirms that the factorization is correct.

Key Concepts

Quadratic Trinomial

A quadratic trinomial is a polynomial consisting of three terms, typically written in the form \(ax^2 + bxy + cy^2\). It's quite common for introductory algebra exercises to require factoring these expressions. The term *quadratic* refers to the highest degree of the variable, which in this form is the square of a variable like \(x^2\). Meanwhile, the "trinomial" part indicates there are three separate terms in the expression.
In the given exercise, the quadratic trinomial is \(8x^2 + 6xy - 27y^2\). Remember:

• \(a = 8\) is the coefficient of \(x^2\),
• \(b = 6\) is the coefficient of \(xy\),
• \(c = -27\) is the coefficient of \(y^2\).

Understanding these parts helps to set up the problem for factoring. Identifying the form is key to knowing which tactic to employ in finding the factors.

Greatest Common Factor

The Greatest Common Factor (GCF) is an essential tool used in simplifying and factoring polynomials. It's the largest number or algebraic expression that divides all the terms in the polynomial without leaving a remainder. In many exercises, it's the first step in the factoring process.
In this exercise, the coefficients of the given expression are \(8\), \(6\), and \(-27\). We assess these coefficients for a GCF. Unfortunately, these numbers do not share any common factors other than 1, indicating that initial factoring trigonometry does not simplify the expression.
Understanding this concept is crucial as recognizing a GCF where possible can significantly ease the process of factoring a trinomial or larger polynomial.

Factor by Grouping

Factoring by grouping is a valuable method when dealing with polynomials, especially when factoring trinomials not easily divisible by a common factor. This technique involves rearranging and grouping terms within the polynomial to reveal common factors in pairs of terms.
For the trinomial \(8x^2 + 6xy - 27y^2\), the task is to split the middle term into two parts that facilitate factoring by grouping. Specifically, the two parts of the middle term should multiply to the product of \(ac\) (coefficients \(a\times c\) in the expression) and add up to \(b\). For this example:

• The product of \(ac = 8 imes (-27) = -216\).
• The sum of the pair should be the middle coefficient: \(6\).
• The numbers fulfilling this are \(18\) and \(-12\).

Hence, we rewrite the expression as two grouped binomials: \((8x^2 + 18xy) + (-12xy - 27y^2)\). Factoring out common factors from each group, we derive two binomials: \((2x - 3y)(4x + 9y)\). Recognizing these steps can enhance your ability to tackle similar problems in polynomial factoring effectively.