Problem 39 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(z^{2}+11 z+24\)

Step-by-Step Solution

Answer

The expression factors as \((z + 3)(z + 8)\).

Step 1: Identify the problem

We want to factor the quadratic expression \(z^{2}+11z+24\). We are looking for two binomials of the form \((z + m)(z + n)\) where \(m\) and \(n\) are integers.

Step 2: Find numbers that satisfy product and sum conditions

To factor \(z^{2}+11z+24\), we need two numbers whose product is 24 (the constant term) and whose sum is 11 (the coefficient of the linear term).

Step 3: Determine the pair of numbers

The pairs of numbers that multiply to 24 are: \((1, 24), (2, 12), (3, 8), (4, 6)\). Among these, \(3\) and \(8\) add up to \(11\).

Step 4: Write the factored form

Using \(3\) and \(8\) as \(m\) and \(n\), the expression \(z^{2}+11z+24\) can be written as \((z + 3)(z + 8)\).

Key Concepts

Binomial Expression

A binomial expression is a polynomial with two distinct terms. It's a simple form of an algebraic expression that you will often encounter in mathematics, especially in algebra. Understanding binomials is crucial since factoring quadratic polynomials involves breaking them down into two binomial expressions.
In a typical binomial, you'll see something like

• (x + m)
• (z - n)

Binomials are fundamental building blocks for many algebraic operations, such as expansion and factorization. When working with quadratics, the goal is typically to rewrite the quadratic equation into a product of two binomials.

Product and Sum Method

The product and sum method is a popular technique for factoring quadratic polynomials. It involves breaking down a quadratic expression into two binomial expressions. Here’s how it works:
To factor using this method, look for two numbers that satisfy two conditions simultaneously:

• The product of these numbers must equal the constant term of the quadratic polynomial.
• The sum of these numbers must equal the coefficient of the linear term in the quadratic polynomial.

For instance, if you have the quadratic polynomial \(z^2 + 11z + 24\), the task is to find numbers whose product is 24 and sum is 11. The numbers 3 and 8 fit perfectly, which allows you to express the quadratic as

• (z + 3)(z + 8)

Understanding this method provides a strong foundation for working with quadratics.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2. Its general form is \(ax^2 + bx + c\), where a, b, and c are constants, and \(a ≠ 0\). Quadratic polynomials are significant in mathematics because they describe a wide range of natural and theoretical phenomena.
When you're asked to factor a quadratic polynomial like \(z^2 + 11z + 24\), you are essentially looking to express it as a product of two simpler polynomials, typically binomials. This process often involves the method of completing the square, using the quadratic formula, or, most commonly for simple cases, the product and sum method.
Knowing how to identify and manipulate quadratic polynomials gives you the tools to solve equations and find the roots, which are the solutions where the polynomial equals zero. This capability is fundamental in fields such as physics, engineering, economics, and biology.