Problem 98 - Algebra for College Students

Question

Suppose that your friend factors \(36 x^{2} y+48 x y^{2}\) as follows: $$ \begin{aligned} 36 x^{2} y+48 x y^{2} &=(4 x y)(9 x+12 y) \\ &=(4 x y)(3)(3 x+4 y) \\ &=12 x y(3 x+4 y) \end{aligned} $$ Is this a correct approach? Would you have any suggestion to offer your friend?

Step-by-Step Solution

Answer

Yes, the approach is correct. No suggestions needed.

Step 1: Identify Common Factors

Observe the expression: \(36x^2y + 48xy^2\). We need to factor out the greatest common factor (GCF) from both terms.

Step 2: Find the Greatest Common Factor

Identify the common factors in the coefficients: 36 and 48, and variables: \(x^2y\) and \(xy^2\).- GCF of 36 and 48 is 12.- For variables, the common factor is \(xy\), as both contain \(x\) and \(y\).- Therefore, the GCF is \(12xy\).

Step 3: Factor out the GCF

Pull the GCF \(12xy\) out of the expression: \[36x^2y + 48xy^2 = 12xy(3x + 4y)\].

Step 4: Confirm the Factored Expression

Check if \(12xy(3x + 4y)\) is equivalent to the original expression by distributing: - Distribute \(12xy\) over \(3x + 4y\): - \(12xy \times 3x = 36x^2y\) and \(12xy \times 4y = 48xy^2\).- Combine: \(36x^2y + 48xy^2\), which confirms the original expression.

Step 5: Evaluate Your Friend's Approach

Your friend’s approach is correct. They correctly identified the greatest common factor and factored it out to obtain \(12xy(3x + 4y)\). No suggestions are necessary.

Key Concepts

Greatest Common Factor (GCF)

When dealing with algebraic expressions, one of the crucial steps is identifying the Greatest Common Factor, or GCF. The GCF is a term or number that can evenly divide each term in the expression.

• For numbers, it involves determining the largest number that divides each of the coefficients without leaving a remainder.
• For variables, you look for the highest power of any common variables present in all the terms.

In this exercise, for example, the expression is \(36x^2y + 48xy^2\). By examining 36 and 48, we find the GCF of these coefficients is 12. Meanwhile, each term has at least one \(x\) and one \(y\), thus the GCF in terms of the variables is \(xy\). So, the overall GCF of the expression is \(12xy\). Recognizing this allows us to simplify and factor expressions neatly.

Distributive Property

The distributive property is your handy companion when you need to expand or factor polynomial expressions. It states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products. The property can be written as:\[ a(b + c) = ab + ac \] Let’s see how it works with our original expression. Once we identify the GCF, \(12xy\), we factor it out, creating the expression \(12xy(3x + 4y)\). By distributing \(12xy\) over \(3x + 4y\), we expand the expression back:

• First, \(12xy \times 3x = 36x^2y\).
• Second, \(12xy \times 4y = 48xy^2\).

This step helps verify that our factored expression, \(12xy(3x + 4y)\), is equivalent to the original, ensuring no steps have resulted in errors.

Polynomial Factoring

Factoring polynomials is a technique used to rewrite an expression as a product of its factors. It's one of the most vital tools in algebra for simplifying expressions and solving equations. The basic idea is to "un-do" the distributive property.Initially, you want to find and factor out the GCF from the entire polynomial. In our case, \(36x^2y + 48xy^2\), the GCF is \(12xy\).Once the GCF is factored out, it's often necessary to check the resulting expression inside the parentheses to see if it can be factored further. For instance, the expression \(3x + 4y\) may sometimes have further factors if it were more complex or contained terms allowing additional factorization.Factoring completely often involves more than just pulling out the GCF. It requires recognizing various factor forms like difference of squares, trinomials, or other common polynomial patterns. However, in this example, \(3x + 4y\) is already in its simplest form, requiring no further action. Factoring polynomials enhances both simplification and understanding of complex mathematical relationships.