Question

Project: Some trinomials that have two variables (such as \(x^{2}+5 x y+6 y^{2}\) ) can be factored by temporarily dropping one variable ( \(y\) in this example), factoring the remaining trinomial \(\left(x^{2}+5 x+6\right)\) into \((x+3)(x+2),\) and then putting back the second variable, getting \((x+3 y)(x+2 y) .\) Try this technique on the following trinomials: $$x^{2}-13 x y+36 y^{2} \quad x^{2}+19 x y+84 y^{2} \quad x^{2}-9 x y+20 y^{2}$$

Short answer

The factored forms are \((x-9y)(x-4y)\), \((x+7y)(x+12y)\), and \((x-5y)(x-4y)\).

Step-by-step solution

Factor the First Trinomial

Temporarily ignore the variable y in the trinomial \(x^{2}-13xy+36y^{2}\). Factor the resulting trinomial \(x^{2}-13x+36\) as if y was not there. Find two numbers that multiply to 36 and add up to -13. These numbers are -9 and -4. Thus, the factored form is \((x-9)(x-4)\). Now reintroduce the variable y to get \((x-9y)(x-4y)\).

Factor the Second Trinomial

Now factor \(x^{2}+19xy+84y^{2}\) using the same technique. Temporarily drop the y and factor \(x^{2}+19x+84\). Find two numbers that multiply to 84 and add up to 19. These numbers are 7 and 12. The factored form is \((x+7)(x+12)\). When we reintroduce y, we get \((x+7y)(x+12y)\).

Factor the Third Trinomial

For the trinomial \(x^{2}-9xy+20y^{2}\), we again temporarily ignore y and factor \(x^{2}-9x+20\). Find two numbers that multiply to 20 and add up to -9, which are -5 and -4. This gives us the factors \((x-5)(x-4)\). Finally, bring back the y to have \((x-5y)(x-4y)\).

Key concepts

Polynomials in Two Variables

Polynomials in two variables, such as those including terms like (x^{2}+5xy+6y^{2}), are algebraic expressions that contain two different variables, often represented by x and y. These expressions can take various forms, but when discussing trinomials, we are referring to a specific type of polynomial that has three terms. Polynomials with two variables can seem daunting at first, but they can often be broken down into more manageable parts through factoring.

Factoring such polynomials is analogous to untangling a complicated knot by carefully finding the right strands to pull. The process allows us to rewrite the polynomial as a product of two or more polynomials, revealing the zeros or solutions of the original polynomial. Understanding the structure of these polynomials lays the groundwork for solving more complex algebraic equations.

Algebraic Factoring Techniques

Algebraic factoring techniques encompass a variety of methods used to simplify polynomials into products of simpler polynomials. This is a fundamental skill in algebra that aids in solving equations and understanding algebraic functions. One common example is factoring by finding common factors, where we look for terms that share a coefficient or variable.

Another technique involves factoring quadratics, (polynomials of degree 2), such as trinomials. This often relies on finding two numbers that can both sum up and multiply to specific values, much like solving a numerical puzzle. For trinomials with two variables, a strategic move is to temporarily reduce the equation to a single variable, simplifying the problem. Once factored, the second variable is reintroduced. The mastery of these methods opens up pathways to solve an array of problems in algebra.

Factoring by Grouping

Factoring by grouping is a technique that can be particularly useful when dealing with polynomials that have four or more terms. To factor by grouping, one usually groups terms with common factors or those that seem to fit together naturally, and then factors out the greatest common factor from each group. This process sometimes reveals a common binomial factor between the groups, allowing the polynomial to be factored further.

Although the textbook problems focus on trinomials, understanding factoring by grouping lays the foundation for tackling more intricate polynomials. It is like solving a jigsaw puzzle, where finding sections that fit together simplifies the larger picture. This technique reinforces the importance of looking for patterns and commonalities within algebraic expressions.