Grade 10 · Grade 10: Polynomials, Models and Function Behaviour · Polynomials, Models and Function Behaviour
Polynomials and Rational Expressions
Operate with polynomials, use zeros to sketch graphs, and simplify rational expressions safely.
Learning objectives
- Add, subtract and multiply polynomials.
- Find zeros from factorisations and sketch polynomial graphs.
- Simplify rational expressions and perform operations with them.
- Solve rational equations, identifying restrictions and extraneous solutions.
AERO Mathematics alignment
AERO.M10.AE.1
Add, subtract and multiply polynomials, and understand that polynomials form a system closed under these operations.
AERO.M10.AE.2
Identify zeros of polynomials when suitable factorisations are available and use the zeros to construct a rough graph of the function.
AERO.M10.AE.3
Simplify rational expressions, perform operations with them, and solve simple rational equations, identifying restrictions and extraneous solutions.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
ENGAGE
Start here
A designer models a curve with y = x³ − 4x.
Think about it
Where does the curve cross the x-axis, and how can factorising tell you before you graph?
Hint: Take out a common factor first.
EXPLAIN
Polynomials and rational expressions
Polynomials are closed under addition, subtraction and multiplication: combining them always produces another polynomial, just as with integers.
Zeros and shape
x³ − 4x = x(x² − 4) = x(x − 2)(x + 2), so the zeros are −2, 0 and 2. Each zero is a crossing point, and knowing them plus the end behaviour gives a reliable rough sketch.
Rational expressions
| Operation | Rule | Watch out for |
|---|---|---|
| Simplify | Cancel common factors | Only factors cancel, never terms |
| Multiply | Multiply numerators and denominators | Factorise first |
| Divide | Multiply by the reciprocal | Exclude zero denominators |
| Add | Use a common denominator | Restrictions still apply |
Restrictions and extraneous solutions
(x + 1)/(x − 3) is undefined at x = 3. When solving rational equations you multiply through by denominators, which can introduce solutions that break the original restriction — always check each answer against the excluded values.
Common misconception
You cannot cancel across a plus sign. In (x + 2)/x the x values do not cancel, because the numerator is a sum, not a product.
INVESTIGATION
Try it yourself
Create a cubic with zeros at −1, 2 and 3 by multiplying its factors. Expand it, sketch the graph and check that your zeros appear where you predicted.
Watch
How to Solve Quadratic Equations by Factoring
The Organic Chemistry Tutor · 10:00
Factorising is how you find zeros and simplify rational expressions.
Before you watch: Why does a factor of (x − 4) give a zero at x = 4?
- Expand (x + 2)(x² − 3x + 1).
- Which value must be excluded from (x + 1)/(x − 3)?
Interactive simulation · PhET
Area Model Algebra
Multiply polynomials with the area model, then read the factors back from the diagram.
While you explore
- How does the model show every partial product?
- Which values must be excluded from a rational expression, and why?
Key vocabulary
- Extraneous solution
- An answer that appears during solving but does not really work.
- Zero of a function
- An input that makes the output zero.
Practice questions
0/1 correct
Level 1 · Criterion A
What are the zeros of y = x³ − 4x?
MYP criterion tasks
Level 2 · Criterion C
Simplify (x² − 9)/(x² + 4x + 3) and state the restrictions.
Reflect & track
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