Problem 20 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 5 . \(8 x^{2}-14 x y+3 y^{2}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Coefficients

Given the trinomial is of the form \(8x^2 - 14xy + 3y^2\). Identify the coefficients: \(a = 8\), \(b = -14\), \(c = 3\). Our goal is to rewrite this trinomial into two binomials.

Step 2: Find Two Numbers with Desired Product and Sum

We need two numbers whose product is \(a \times c = 8 \times 3 = 24\) and whose sum is the coefficient \(b = -14\). The numbers satisfying these conditions are \(-12\) and \(-2\).

Step 3: Rewrite the Middle Term

Rewrite the middle term, \(-14xy\), as \(-12xy - 2xy\). The trinomial becomes \(8x^2 - 12xy - 2xy + 3y^2\).

Step 4: Group and Factor by Grouping

Now group the terms: \((8x^2 - 12xy) - (2xy - 3y^2)\). Factor out the greatest common factor in each group. The expression becomes \(4x(2x - 3y) - y(2x - 3y)\).

Step 5: Factor by Common Binomial

Notice that \((2x - 3y)\) is common in both groups. Factor \((2x - 3y)\) out: \((4x - y)(2x - 3y)\). This is the factored form of the original trinomial.

Key Concepts

Polynomial Coefficients

In polynomials, coefficients are the numbers in front of the variables that define the specific structure of the polynomial. When dealing with a trinomial like \(8x^2 - 14xy + 3y^2\), the coefficients are essential in helping to factor the expression.
Let's break it down:

• \(a = 8\): This is the coefficient of the \(x^2\) term.
• \(b = -14\): This is the coefficient for the mixed term \(xy\).
• \(c = 3\): This is the coefficient of the \(y^2\) term.

Understanding how to identify and use these coefficients is vital when rewriting the polynomial into a factorable form. Without them, it would be challenging to determine relationships between terms necessary for factoring.

Factoring by Grouping

Factoring by grouping is a method used to simplify complex polynomials into more manageable expressions.
Here's how it works:

• Break the polynomial into two groups. For \(8x^2 - 14xy + 3y^2\), we reshaped it into \((8x^2 - 12xy)\) and \((-2xy + 3y^2)\).
• Identify and factor out the greatest common factor from each group.

For instance:

• In the group \(8x^2 - 12xy\), the GCF is \(4x\), giving us \(4x(2x - 3y)\).
• For \(-2xy + 3y^2\), the GCF is \(-y\), simplifying it to \(-y(2x - 3y)\).

This method leads directly to a simpler polynomial, making it easier to identify common factors in subsequent steps.

Binomial Factors

Binomial factors are pairs in which trinomials or polynomials are expressed. In our equation, upon achieving the step-by-step simplification and after grouping, we can observe:

• The expression was rewritten as \(4x(2x - 3y) - y(2x - 3y)\).
• The binomial \((2x - 3y)\) emerges as a common factor across both terms.

This allows us to take out the common binomial, arriving at the final factored form, \((4x - y)(2x - 3y)\). Finding these factors is critical, as it simplifies the trinomial into a product of two binomials, making it more understandable and solvable.

Product and Sum Method

The product and sum method is a strategy for factoring trinomials, especially those that seem complicated at first glance. Here's the step-by-step rundown:

• Calculate the product of \(a\) and \(c\). In our case, it’s \(8 \times 3 = 24\).
• Identify two numbers that have this product (24) and sum to \(b = -14\). Here, the numbers are \(-12\) and \(-2\).

By rewriting the middle term \(-14xy\) with these two numbers as \(-12xy - 2xy\), we make the trinomial ready for grouping.
This process is powerful because it converts a complex expression into manageable parts, ready for the factoring by grouping method. It's a foundational technique in algebra, leading to quick and clear solutions.