Problem 15 - Algebra 2

Question

Factor each expression. $$ r^{2}-11 r+18 $$

Step-by-Step Solution

Answer

\((r - 9)(r - 2)\)

Step 1: Find two numbers that multiply to the constant term and add to the linear coefficient

To factor the quadratic expression, we need two numbers that multiply together to give the product of the constant term, 18, and at the same time add up to the linear coefficient, which is -11.

Step 2: Determine the numbers

The numbers we are looking for are -9 and -2 because (-9) * (-2) = 18 and (-9) + (-2) = -11.

Step 3: Rewrite the quadratic in its factored form

Using the numbers found in step 2, rewrite the quadratic expression as \((r - 9)(r - 2)\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental aspect of algebra that students must master. In essence, they take the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). These equations are called 'quadratic' because the highest power of the variable \((x^2\)) is 2, which is also known as a square term.

When solving quadratic equations by factoring, we aim to break them down into simpler binomial expressions that can be multiplied together to give the original equation. The core idea is to find two numbers that can both add up to the linear coefficient \(b\) and multiply together to get the constant term \(c\), just like in our exercise example. This is often referred to as 'finding the roots' or 'solving for \((x\)).' The factored form of a quadratic equation can be used to find its solutions, which are the values of \(x\) that satisfy the equation when set to zero.

Polynomial Factoring

Polynomial factoring is the process of breaking down a polynomial into a product of simpler polynomials. This can include the factoring of quadratic expressions as we saw earlier, but can also apply to polynomials of higher degrees. The goal here is to express the polynomial as a product of factors that cannot be factored further, known as irreducible factors.

It's a bit like finding the prime factors of a number but instead applied to algebraic expressions. Key strategies in polynomial factoring include looking for a common factor in all the terms, utilizing the difference of squares, and the sum or difference of cubes. In practice, it becomes essential to recognize patterns and use different techniques depending on the nature of the polynomial. Factoring is a critical skill in algebra because it simplifies expressions and makes other operations, like solving equations, much easier.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) that represent a particular quantity. Unlike equations, they do not have an equals sign and do not express a relationship between two things. Instead, they are like phrases or sentences in the language of mathematics that can be manipulated and transformed according to algebraic rules and properties.

The expression \(r^2 - 11r + 18\) from our exercise is a prime example of an algebraic expression. It's a polynomial because it involves several terms that are summed or subtracted. Understanding how to manage these expressions is crucial to students' success in algebra. Operations like expansion, factoring, simplification, and evaluation are all part of working with algebraic expressions. Learning to factor algebraic expressions is like learning to decompose a complex idea into simpler, more manageable pieces, making it a vital skill for problem-solving in mathematics.