Problem 14 - Elementary Algebra

Question

Factor completely. Identify any prime polynomials. $$ 63 p w^{2}+18 p w+231 m w^{2}+66 m w $$

Step-by-Step Solution

Answer

3w(7w + 2)(3p + 11m)

Step 1: Identify the Greatest Common Factor (GCF)

First, identify the GCF of all the terms in the polynomial. The terms are: 63pw^{2}, 18pw, 231mw^{2}, and 66mw. The GCF for the coefficients 63, 18, 231, and 66 is 3. Each term also contains either p or m and w as factors. Therefore, the GCF for the entire polynomial is 3w. Factoring out 3w gives: 3w(21pw + 6p + 77mw + 22m).

Step 2: Factor by Grouping

Group the terms inside the parenthesis in pairs to factor them further: 3w[(21pw + 6p) + (77mw + 22m)]. Now factor out the common factors within each group: 3w[3p(7w + 2) + 11m(7w + 2)].

Step 3: Factor the Common Binomial

Observe that (7w + 2) is a common binomial factor in both groups. Factor (7w + 2) out: 3w(7w + 2)(3p + 11m).

Key Concepts

Greatest Common Factor (GCF)

Understanding the Greatest Common Factor (GCF) is essential in simplifying polynomials. It represents the largest factor that divides all terms in an algebraic expression without leaving a remainder.
In our exercise, the polynomial is: \(63pw^2 + 18pw + 231mw^2 + 66mw\).
Here are the steps to find the GCF:

• First, list the coefficients: 63, 18, 231, and 66.
• Find the GCF of these numbers. The GCF is 3.
• Next, look at the variables: each term contains 'w', and some contain 'p' or 'm.' The common variable factor is 'w.'
• Thus, the GCF for the entire polynomial is 3w.

Now factor out this GCF from the polynomial:
3w(21pw + 6p + 77mw + 22m).
By factoring out the GCF first, we simplify the polynomial and make the task of further factoring much easier.

Factor by Grouping

After finding the GCF and simplifying the polynomial, we use a method called 'Factor by Grouping.' This method is particularly useful for polynomials with four or more terms.
In our simplified polynomial: 3w(21pw + 6p + 77mw + 22m), we can group the terms into pairs for easier factoring:
3w[(21pw + 6p) + (77mw + 22m)].
Next, factor out the common factors in each group:

• In the first group (21pw + 6p), the common factor is 3p, giving us 3p(7w + 2).
• In the second group (77mw + 22m), the common factor is 11m, giving us 11m(7w + 2).

Now our expression looks like this:
3w[3p(7w + 2) + 11m(7w + 2)].
By grouping terms and factoring out common factors within them, we prepare the polynomial for the final step: factoring out the common binomial.

Algebraic Expressions

Algebraic expressions, such as polynomials, consist of variables, coefficients, and arithmetic operations. They can represent a wide range of mathematical problems.
In our exercise, the initial algebraic expression was: \(63pw^2 + 18pw + 231mw^2 + 66mw\). Through careful analysis and factoring, we simplified and grouped this polynomial.

• We factored out the GCF of 3w.
• We further simplified by grouping terms and factoring out common factors within those groups.
• Finally, we identified a common binomial factor in the grouped expression.

These steps transformed our expression into something more manageable:
3w(7w + 2)(3p + 11m).
Understanding these manipulations is key to mastering algebra and solving polynomial equations. Each step involves recognizing patterns and applying factorization techniques to simplify and solve algebraic expressions effectively.