Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 121 r^{3} s^{4}+77 r^{2} s^{4}-55 r^{4} s^{3} $$

Step-by-Step Solution

Answer

The factored form is \(11r^2s^3(11rs + 7s - 5r^2)\).

Step 1: Identify the Greatest Common Factor (GCF)

First, you need to identify the greatest common factor (GCF) for all the terms in the polynomial. Look at the numbers 121, 77, and -55; their GCF is 11. Next, look at the variables: each term has common factors of at least \(r^2\) and \(s^3\). Thus, the GCF is \(11r^2s^3\).

Step 2: Factor Out the GCF

Now that you have identified the GCF as \(11r^2s^3\), factor it out from each term in the polynomial:\[ 11r^2s^3(11rs + 7s - 5r^2) \].This is achieved by dividing each term by \(11r^2s^3\) and rewriting the expression.

Step 3: Simplification and Rewriting

Ensure that the expression inside the parentheses is correct. The polynomial \(11rs + 7s - 5r^2\) is what remains after factoring out \(11r^2s^3\). This cannot be factored further with integer coefficients.

Key Concepts

greatest common factor

When dealing with polynomials, one of the first steps to simplifying them is to find the Greatest Common Factor (GCF). The GCF is the largest factor that divides all the terms of the polynomial. This factor can include both numbers and variables.

For example, consider the terms in the expression: 121, 77, and -55. The largest number that can divide all of these is 11.

- Numerical GCF: Here, the GCF of the coefficients 121, 77, and -55 is 11.

- Variable GCF: Look at the commonality in the variables. For instance, if each term has some power of 'r' and 's', consider the smallest power among them. In this case, the common variable factors are at least \(r^2\) and \(s^3\).

Once the GCF is identified, it can be factored out from the polynomial, making further operations simpler and tidier.

factoring polynomials

Factoring polynomials involves expressing it as a product of its factors. This helps in simplifying polynomials, solving equations, and understanding them better.

In the given expression \(121r^3s^4+77r^2s^4-55r^4s^3\), the process started with identifying the GCF of all terms, which was \(11r^2s^3\).

- **Extracting the GCF**: By dividing each term by this GCF and rewriting the expression, it reveals the potential remaining factor, turning the polynomial into a simplified expression: \(11r^2s^3(11rs + 7s - 5r^2)\).

Basic factorization involves such straightforward steps but understanding deeper types may involve patterns like the difference of squares or special polynomials like quadratic trinomials.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition or multiplication). They form the basis of algebra. Understanding how to manipulate these expressions is essential in solving algebraic problems.

When we work with polynomial expressions (like \(121r^3s^4+77r^2s^4-55r^4s^3\)), we use rules of algebra to rearrange or simplify them.

- **Simplicity**: To simplify, always look for common factors or special patterns.

- **Terms and coefficients**: Recognize individual components, like the coefficients (numbers before variables), and manipulate them using algebraic laws.

By integrating the concept of the GCF and polynomial factors, this creates a more manageable and simplified form which is easier to evaluate or solve.