Problem 50 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$6 x^{2}-7 x y-5 y^{2}$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(6x^2-7xy-5y^2\) is (2x + y)(3x - 5y)

Step 1: Identify the Coefficients and Constants

First, identify the coefficients and the constant of the trinomial. In \(6x^2-7xy-5y^2\), the coefficients of \(x^2\) and \(y^2\) are 6 and -5, respectively, and the coefficient of \(xy\) is -7.

Step 2: List all pairs of factors

List all pairs of factors for the product of the coefficient of \(x^2\) and the coefficient of \(y^2\), which is \(6 * -5 = -30\). The pairs are \((-1, 30),(1, -30),(-2, 15),(2, -15),(-3, 10),(3, -10),(-5, 6),(5, -6),(-6, 5),(6, -5),(-10, 3),(10, -3),(-15, 2),(15, -2),(-30, 1),(30, -1)\)

Step 3: Look for a pair of factors that sum to the coefficient

Look for a pair of factors from the one generated above that adds up to the coefficient of the middle term, -7. The pair that satisfies this condition is (-10, 3)

Step 4: Rewrite the middle term

Replace the middle term (-7xy) of the trinomial with -10xy + 3xy. The trinomial becomes \(6x^2 - 10xy + 3xy - 5y^2\)

Step 5: Factor by grouping

Group the terms two-by-two and factor out the greatest common factor (GCF) from each pair. This results in \(2x(3x - 5y) + y(3x - 5y)\)

Step 6: Factor out the common binomial

Since \(3x - 5y\) is a common factor of each pair, factor it out to get (2x + y)(3x - 5y)

Step 7: Check Factoring using FOIL

Now multiply the factored form using the FOIL method to confirm if it's the same as the original trinomial given in the problem. The product using FOIL is indeed \(6x^2-7xy-5y^2\) confirming the factored form is correct

Key Concepts

Coefficient Identification

To start factoring any trinomial, we need to identify specific terms including coefficients and constants involved in the equation. For the provided trinomial, \(6x^2 - 7xy - 5y^2\), this involves understanding what 'coefficients' are. Coefficients are numbers that multiply the variables in an expression. In our example:

• The coefficient of \(x^2\) is 6.
• The coefficient of \(xy\) is -7.
• The coefficient of \(y^2\) is -5.

Recognizing these coefficients is crucial as they guide us in the next steps of the problem. They are your keys to break down the equation and discover potential factors that simplify the trinomial. By balancing and rearranging these coefficients properly, we aim to transform the expression into a product of simpler polynomials.

Factor Pairs

Once coefficients are identified, the next job is to explore the potential factor pairs. In this method, we calculate the product of the first and last coefficients in our trinomial, \(6\) and \(-5\), which equals \(-30\). We are looking for two numbers that multiply to this product \(-30\) but also add up to the middle coefficient, \(-7\). Listing these pairs helps evaluate which fits best:

• Pairs like \((3, -10)\) and \((-10, 3)\).
• Others include \((5, -6)\) and \((-6, 5)\).

Choosing the correct pair is key in ensuring that the equation balances properly. For our trinomial, the pair \((3, -10)\) correctly adds to \(-7\). Thus, this step involves a bit of trial and error, but logical deduction will lead you to the right numbers.

Factor by Grouping

After identifying the correct factor pair, we proceed to rewrite and factor the expression by grouping. It involves splitting the middle term using the selected factor pair. For \(6x^2 - 7xy - 5y^2\), we rewrite it using \(-10xy\) and \(3xy\):
\(6x^2 - 10xy + 3xy - 5y^2\). This alteration enables us to group terms logically:

• First Group: \(6x^2 - 10xy\).
• Second Group: \(3xy - 5y^2\).

By factoring each group separately, we pull out the greatest common factor from each:

• From \(6x^2 - 10xy\), we factor out \(2x\) giving us \(2x(3x - 5y)\).
• From \(3xy - 5y^2\), we factor out \(y\) giving us \(y(3x - 5y)\).

Notice the common binomial factor \((3x - 5y)\) in both terms. This allows us to further simplify by factoring out \((3x - 5y)\) from the whole expression:
\((2x + y)(3x - 5y)\).
Thus, through factoring by grouping, we effectively reduce the trinomial into simpler binomial factors.