Problem 66 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(42 x^{4}-99 x^{3} y-15 x^{2} y^{2}\)

Step-by-Step Solution

Answer

Factor the trinomial as \(3x^2(7x + y)(2x - 5y)\).

Step 1: Identify Common Factors

Examine the trinomial for any common factors among the terms. All terms have a common factor of \(3x^2\).

Step 2: Factor Out the Common Factor

Factor \(3x^2\) from each term of the trinomial. This gives you: \[3x^2 (14x^2 - 33xy - 5y^2)\].

Step 3: Factor the Resulting Trinomial

Now, focus on the trinomial \(14x^2 - 33xy - 5y^2\). Look for two numbers that multiply to \(14 \times (-5) = -70\) and add to \(-33\). These numbers are \(-35\) and \(2\). Rewrite \(-33xy\) using these two numbers: \[14x^2 - 35xy + 2xy - 5y^2\].

Step 4: Group Terms and Factor by Grouping

Group the terms and factor each group:\((14x^2 - 35xy) + (2xy - 5y^2)\).Factor each group:\[7x(2x - 5y) + y(2x - 5y)\].

Step 5: Factor Out the Common Binomial Factor

Notice that both groups contain \(2x - 5y\) as a common factor. Factor this out:\[(7x + y)(2x - 5y)\].

Step 6: Combine Everything

Put back the common factor \(3x^2\) to the trinomial factors obtained to get the fully factored form:\[3x^2(7x + y)(2x - 5y)\].

Key Concepts

Trinomial

A trinomial is a type of polynomial that consists of three terms. Think of it as a mathematical expression that looks like this: three sets of numbers and variables combined, each separated by a plus or minus sign. In the exercise, the trinomial given was \(42x^4 - 99x^3y - 15x^2y^2\). This type of polynomial often appears in algebra, where the goal is typically to factor or simplify it. Understanding trinomials is essential for solving many algebraic problems.
Trinomials can often be factored or decomposed into simpler expressions. Factoring is the process of breaking down these expressions into products of other polynomials. A trinomial, like any polynomial, comes with variables (often represented by letters like \(x\) or \(y\)) and coefficients (numbers that are multiplied by the variables). These play an essential role when we need to simplify or manipulate the expression. By mastering trinomials, you open the door to advanced algebraic operations.

Factoring by Grouping

Factoring by grouping is a handy technique used when a polynomial can be split into groups of terms that share a common factor. This method isn't as direct as simple factoring, but it's very effective for complex polynomials. Let's break it down using our trinomial. Once the common factor \(3x^2\) is factored out, we get the expression \(14x^2 - 33xy - 5y^2\).
To factor by grouping, we rearrange and divide this expression into two groups: \((14x^2 - 35xy)\) and \((2xy - 5y^2)\). Each of these groups can then be factored separately. For the first group \((14x^2 - 35xy)\), we factor out \(7x\), leaving \((2x - 5y)\). For the second group \((2xy - 5y^2)\), we factor out \(y\), also leaving \((2x - 5y)\). The success here lies in creating a common factor in both groups.

Common Factors

Finding common factors is the first step in factoring any polynomial, including trinomials. A common factor is a factor that divides each term in the polynomial without leaving a remainder. In the problem given, the common factor was \(3x^2\). By identifying this, the expression was simplified from \(42x^4 - 99x^3y - 15x^2y^2\) to \(3x^2(14x^2 - 33xy - 5y^2)\).
Recognizing common factors simplifies more complex problems, making them easier to manage. It reduces the polynomial into simpler components, setting the stage for additional factoring methods like grouping. To find the common factors, examine the numerical coefficients and variable terms in each part of the polynomial. Be systematic in your approach—start with the smallest numbers or simple terms. This strategy ensures you effectively break down the polynomial into its simplest form, a crucial skill in algebra.

Binomial Factors

A binomial is a polynomial with exactly two terms. Once you have used factoring by grouping, you often end up with binomial factors that share a common element. In the solution provided, after grouping, both parts of the expression had \((2x - 5y)\) as a common binomial factor. This allowed the entire group to be rewritten as \((7x + y)(2x - 5y)\).
Identifying binomial factors involves noticing patterns within the polynomial. Reinforcing key algebraic concepts, binomial factors simplify calculations and allow for the further breakdown of complex expressions. When you spot a matching pattern or pair, like \((2x - 5y)\) in this case, factor it out to achieve a more concise form. This not only makes the polynomial easier to manipulate but also reveals relationships between terms that you may use later in solving equations. Binomial factors are fundamental building blocks that simplify algebraic tasks.