Problem 35 - Elementary Algebra

Question

Factor completely. $$ 4 x 2-19 x y+12 y 2 $$

Step-by-Step Solution

Answer

The expression factors to \((4x - 3y)(x - 4y)\).

Step 1: Identify the Form

The expression is a quadratic trinomial in the form of a quadratic expression in two variables, specifically \(4x^2 - 19xy + 12y^2\). Factorization can be approached by looking for two binomials whose product equals the original expression.

Step 2: Find the Product and Sum

In a trinomial \(ax^2 + bxy + cy^2\), we need to find two numbers that multiply to \(a*c\) (the product of the first and last coefficients) and add to \(b\) (the middle coefficient). In this case, \(a = 4\), \(b = -19\), and \(c = 12\), so \(a*c = 48\). We are looking for two numbers that multiply to 48 and add to -19.

Step 3: Determine the Pair

The two numbers that satisfy the conditions are -3 and -16 because \((-3) * (-16) = 48\) and \((-3) + (-16) = -19\). These numbers will help break down the middle term.

Step 4: Rewrite the Expression

Rewrite the middle term of the expression \(-19xy\) using our two numbers found in the previous step: \(4x^2 - 3xy - 16xy + 12y^2\). This splits the middle term into two parts, allowing for grouping.

Step 5: Apply Grouping

Group the terms into two pairs: \((4x^2 - 3xy)\) and \((-16xy + 12y^2)\). This will allow us to factor by grouping.

Step 6: Factor Each Group

For the first group \((4x^2 - 3xy)\), factor out \(x\) to get \(x(4x - 3y)\). For the second group \((-16xy + 12y^2)\), factor out \(-4y\) to get \(-4y(4x - 3y)\).

Step 7: Combine Factors

Notice that \(4x - 3y\) is a common factor in both groups. Factor this out to get \((4x - 3y)(x - 4y)\). The expression is now completely factored.

Key Concepts

Quadratic trinomial

A quadratic trinomial is a polynomial with three terms and involves squares of variables. In our case, the quadratic trinomial is given as \(4x^2 - 19xy + 12y^2\). This expression showcases two variables \(x\) and \(y\), and is characterized by the presence of a term with \(x^2\) and another with \(y^2\).
Quadratic trinomials of this type often follow the pattern \(ax^2 + bxy + cy^2\), where \(a\), \(b\), and \(c\) are constants. These expressions can sometimes seem daunting due to their complexity. However, they can be simplified and factored using specific strategies, which leads us to the final goal of completely factoring the trinomial. By breaking down the quadratic trinomial, we can make it easier for us to manipulate and work with.

Binomial product

A binomial product involves two binomials multiplied together. When factored, a quadratic trinomial becomes a product of two binomials. For example, when we factor \(4x^2 - 19xy + 12y^2\), it can be expressed as a product \((4x - 3y)(x - 4y)\).
Understanding binomial products is essential because it connects directly to the original trinomial. This transformation is crucial as it reveals the roots and solutions of the quadratic equation. By mastering the ability to express polynomials as binomial products, you gain a powerful tool that simplifies understanding and solving polynomial equations.
To achieve this, we need to explore the relationships between coefficients in the trinomial and the terms in the factors. Therefore, we can relate the binomials to \(ax^2 + bxy + cy^2\), providing clarity and straightforwardness in handling quadratic expressions.

Grouping method

The grouping method is a handy technique for factoring polynomials, specifically useful for expressions like quadratic trinomials. This method involves breaking the middle term into two parts based on two numbers that satisfy specific conditions: they must multiply to give \(a*c\) and add up to \(b\).
In the case of \(4x^2 - 19xy + 12y^2\), we needed numbers that multiply to \(48\) (since \(a = 4\) and \(c = 12\), thus \(a*c = 48\)) and add to \(-19\). The numbers \(-3\) and \(-16\) fit perfectly as they both multiply to \(48\) and sum to \(-19\).
Once you have these numbers, the method involves rewriting the middle term, followed by grouping parts of the expression: \(4x^2 - 3xy - 16xy + 12y^2\). From there, you factor out the greatest common factors from each group, leading to a common binomial factor. This method streamlines the process, breaking down complex expressions into manageable pieces, making factorization simpler and more predictable.