Solving Problems with Polynomial Functions

Solve polynomial equations and inequalities algebraically and graphically. Grade 12.

Lesson 2.6 of Polynomial Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Introduction to Polynomial Applications. Discover why polynomials matter in real life! Learn a systematic 6-step process for solving application problems with a detailed consecutive integers example.
• Cube Root Applications. Apply cube roots to solve a real-world problem about a broken pole. Learn how to set up equations involving cube roots and solve them using polynomial techniques.

Inside the lesson — a free preview

Why Do We Use Polynomials in Real Life?

Polynomials aren't just abstract math problems - they appear everywhere in the real world! From designing boxes to calculating volumes, understanding motion to optimizing production, polynomial equations help us solve practical problems.

The Problem-Solving Process

When solving application problems with polynomials, we follow a systematic approach:

Step 1: Read and understand.

Step 2: Define variables.

Step 3: Write an equation.

Step 4: Solve the equation.

Step 5: Check for reasonableness.

Step 6: Answer the question.

Step 1: Understand the problem

We need to find three numbers that:.

Step 2: Define the variable

Let x = the first integer.

Why?. Consecutive integers differ by 1, so if the first is x, the next is x+1, and the one after that is x+2.

Step 3: Write the equation

The product of these three numbers equals 24:

The equation. x(x + 1)(x + 2) = 24

Step 4: Expand and simplify

First, multiply the first two factors. x(x + 1) = x^2 + x

Now multiply by the third factor. (x^2 + x)(x + 2)

Distribute. = x^2(x + 2) + x(x + 2)

= x^3 + 2x^2 + x^2 + 2x

= x^3 + 3x^2 + 2x

So our equation is. x^3 + 3x^2 + 2x = 24

Step 5: Set equal to zero

Move 24 to the left side:

Standard form. x^3 + 3x^2 + 2x - 24 = 0

Step 6: Solve using factoring

This is a cubic equation. Let's try to find a rational root.

Using Integral Zero Theorem (leading coefficient is 1). Possible integer roots: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24

Test x = 2. f(2) = (2)^3 + 3(2)^2 + 2(2) - 24

f(2) = 8 + 12 + 4 - 24

f(2) = 0 ✓

Found one!. x = 2 is a solution

Worked example: Detailed Example 1: Consecutive Integers

Question. Three consecutive positive integers have a product of 24. What are the numbers?

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