Problem 35 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 13 x^{2} y^{5} c-26 x^{2} y^{5} c-39 x^{2} y^{5} $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial is \(13x^{2}y^{5}(c - 2c - 3)\).

Step 1: Identify the Greatest Common Factor (GCF)

The given polynomial is: $$ 13x^{2}y^{5}c - 26x^{2}y^{5}c - 39x^{2}y^{5} $$ To find the GCF of the terms, we need to look at common variables and their corresponding powers. In our case, all terms have the variables \(x\) and \(y\) in common. Now we need to find the lowest powers of these variables present in all terms. The lowest power of \(x\) is \(x^{2}\) and the lowest power of \(y\) is \(y^{5}\). The GCF also includes common coefficients like 13 in this case. So, the GCF is \(13x^{2}y^{5}\).

Step 2: Factor the polynomial using GCF

Now we will factor out the GCF from the polynomial. We want to find what remains when we factor out the GCF from each term. So we divide each term by the GCF: $$ \frac{13x^{2}y^{5}c}{13x^{2}y^{5}} = c $$ $$ \frac{-26x^{2}y^{5}c}{13x^{2}y^{5}} = -2c $$ $$ \frac{-39x^{2}y^{5}}{13x^{2}y^{5}} = -3 $$ Now the factored polynomial is: $$ 13x^{2}y^{5}(c - 2c - 3) $$

Key Concepts

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the highest number or expression that divides into two or more numbers or algebraic expressions without leaving a remainder. In the context of factoring polynomials, the GCF is crucial because it simplifies polynomials into more manageable pieces.

Here's how to find the GCF in a polynomial: List the factors of each term separately, then identify the common factors across all terms. Among these, the greatest factor is the GCF. It's important to include both numeric coefficients and variable factors. For example, in the polynomial given, the GCF is determined by identifying the smallest exponent of each common variable in all terms, as well as any common numerical factors. In this case, the GCF is \(13x^{2}y^{5}\).

Once identified, you factor the GCF out of the polynomial, which greatly simplifies the expression and aids in solving or further manipulation.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation (e.g., addition, subtraction, multiplication, division). When dealing with algebraic expressions in polynomials, we often perform operations like factoring to simplify them or solve equations.

To factor an algebraic expression, start by identifying terms that have common factors. This can include both numerical coefficients and variables. Then, use the distributive property to 'pull out' these common factors, which simplifies the expression. For instance, in our exercise, after finding the GCF \(13x^{2}y^{5}\), we use it to factor the polynomial, creating a simplified expression where the GCF is multiplied by the remaining terms.

Variable Exponents

Variable exponents indicate how many times a variable is multiplied by itself. They play a key role in algebra when it comes to finding the GCF of a polynomial. To factor correctly, it's necessary to understand that when multiple terms share variables raised to powers, the lowest exponent of each variable common to all terms is used in the GCF.

Let's look at the term \(x^{2}y^{5}\) from our exercise. The exponent 2 on \(x\) tells us that \(x\) is used twice as \(x \times x\), and the exponent 5 on \(y\) tells us that \(y\) is used five times as \(y \times y \times y \times y \times y\). The GCF uses the lowest exponents because factoring it out must leave whole number exponents in the remaining expression. Reducing variable exponents via factoring helps to simplify expressions and solve equations involving variables.