Problem 128 - Elementary Algebra

Question

Factor each expression. \(r^{2}+11 r s+36 s^{2}\)

Step-by-Step Solution

Answer

The factored expression is (r + 4s)(r + 9s).

Step 1: Identify coefficients and constants

The given expression is a quadratic trinomial of the form ax^2 + bx + c. Here, identify: a = 1 b = 11s c = 36s^2

Step 2: Find two numbers

Find two numbers that multiply to the constant term 36s^2 and add up to the middle coefficient, 11s. After some checks, the two numbers are 9s and 4s.

Step 3: Rewrite the middle term

Rewrite the middle term of the quadratic trinomial using the two numbers found. This gives: r^2 + 9rs + 4rs + 36s^2.

Step 4: Factor by grouping

Group the terms into pairs and factor each pair: r(r + 9s) + 4s(r + 9s).

Step 5: Combine like terms

Factor out the common binomial factor (r + 9s) to get (r + 4s)(r + 9s).

Key Concepts

quadratic equations

Quadratic equations show up all around us. They're unique because they involve a squared term. You might see them in the format \( ax^2 + bx + c = 0 \). In our exercise, the quadratic trinomial is \( r^2 + 11rs + 36s^2 \).

The key to solving quadratic equations is identifying the coefficients and constant terms. Here, \( a = 1 \), \( b = 11s \), and \( c = 36s^2 \).

The solutions to quadratic equations can reveal the points where a parabola, representing the equation, crosses the x-axis.

To solve by factoring, you need to find either the roots or the factors that simplify the equation.

factoring by grouping

Factoring by grouping is a powerful method for handling polynomial expressions. After identifying the numbers 9s and 4s, which multiply to our constant term \(36s^2\) and add up to \(11s\), we rewrite the middle term.

This transforms our equation from \( r^2 + 11rs + 36s^2 \) into \( r^2 + 9rs + 4rs + 36s^2 \).

Now, we group the first two terms together and the last two terms together: \( (r^2 + 9rs) + (4rs + 36s^2) \).

Next, factor each pair separately: \( r(r + 9s) + 4s(r + 9s) \). Finally, you'll see a common binomial factor, \( (r + 9s) \). When combined, it results in \( (r + 4s)(r + 9s) \).

This method simplifies the polynomial by packaging like terms together.

algebraic expressions

Algebraic expressions consist of variables, constants, and operations. They can seem complex, but breaking them down is key.

In our exercise, the given expression is \( r^2 + 11rs + 36s^2 \), containing r's and s's with multiplication and addition.

Recognizing the structure lets you apply techniques like factoring by grouping.

By rewriting expressions as \( r^2 + 9rs + 4rs + 36s^2 \), it’s clear how components relate.

When you break down algebraic expressions, start by identifying parts like coefficients \( (1, 11s, 36s^2) \).

Follow with finding pairs to combine and simplify the problem. Understanding terms and their roles can make algebra less intimidating.