Polynomial Functions
Understand degree, leading coefficient, end behaviour, and the factor and remainder theorems for polynomials.
What is a Polynomial?
A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer powers of the variable.
P(x) = anxn + an-1xn-1 + ... + a1x + a0
where an ≠ 0 and n is a non-negative integer
| Degree | Name | Example | Max Turning Points |
|---|---|---|---|
| 0 | Constant | y = 5 | 0 |
| 1 | Linear | y = 2x + 1 | 0 |
| 2 | Quadratic | y = x² - 3x + 2 | 1 |
| 3 | Cubic | y = x³ - x | 2 |
| 4 | Quartic | y = x⁴ - 5x² + 4 | 3 |
End Behaviour
The end behaviour of a polynomial describes what happens to y as x approaches positive or negative infinity. It depends on two things: the degree and the sign of the leading coefficient.
Even Degree (n = 2, 4, 6, ...)
a > 0: Both ends rise (up-up)
a < 0: Both ends fall (down-down)
Odd Degree (n = 1, 3, 5, ...)
a > 0: Falls left, rises right (down-up)
a < 0: Rises left, falls right (up-down)
Factor and Remainder Theorems
Factor Theorem
If P(a) = 0, then (x - a) is a factor of P(x).
Conversely, if (x - a) is a factor, then P(a) = 0.
Remainder Theorem
When P(x) is divided by (x - a), the remainder is P(a).
This allows you to find remainders without performing long division.
To fully factorise a polynomial: (1) use the factor theorem to find a root, (2) perform polynomial long division or synthetic division to reduce the degree, (3) repeat until fully factored.
Key Vocabulary
Degree
The highest power of the variable in the polynomial. It determines the general shape and maximum number of roots.
Leading Coefficient
The coefficient of the highest-degree term. Along with degree, it determines end behaviour.
End Behaviour
The behaviour of the graph as x approaches positive or negative infinity.
Root (Zero)
A value of x for which P(x) = 0. Corresponds to an x-intercept on the graph.
Worked Examples
State the degree, leading coefficient, and end behaviour of P(x) = -2x⁴ + 3x³ - x + 5.
Step 1: The highest power is 4, so the degree is 4 (quartic).
Step 2: The leading coefficient is -2.
Step 3: Even degree with negative leading coefficient: both ends fall.
Answer: Degree 4, leading coefficient -2. As x → ±∞, y → -∞ (both ends point downward).
Find the remainder when P(x) = x³ + 3x² - 2x + 1 is divided by (x - 2).
Step 1: By the Remainder Theorem, the remainder is P(2).
Step 2: P(2) = (2)³ + 3(2)² - 2(2) + 1 = 8 + 12 - 4 + 1 = 17.
Answer: The remainder is 17.
Show that (x + 1) is a factor of P(x) = 2x³ + 5x² + x - 2, and hence factorise completely.
Step 1: P(-1) = 2(-1)³ + 5(-1)² + (-1) - 2 = -2 + 5 - 1 - 2 = 0. So (x + 1) is a factor.
Step 2: Divide: (2x³ + 5x² + x - 2) ÷ (x + 1) = 2x² + 3x - 2.
Step 3: Factor the quadratic: 2x² + 3x - 2 = (2x - 1)(x + 2).
Answer: P(x) = (x + 1)(2x - 1)(x + 2).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the degree of P(x) = 5x³ - 2x⁴ + 7x - 1?
Question 2
What is the remainder when P(x) = x³ - 4x + 5 is divided by (x - 1)?
Question 3
A degree-5 polynomial with a positive leading coefficient has end behaviour:
Question 4
What is the maximum number of x-intercepts a degree-4 polynomial can have?
Question 5
If (x - 3) is a factor of P(x), then P(3) equals:
Key Concepts Summary
- ●The degree is the highest power and determines the maximum number of roots and turning points.
- ●A degree-n polynomial has at most n real roots and at most n - 1 turning points.
- ●End behaviour depends on the degree (even/odd) and the sign of the leading coefficient.
- ●The Factor Theorem: P(a) = 0 implies (x - a) is a factor of P(x).
- ●The Remainder Theorem: dividing P(x) by (x - a) gives a remainder of P(a).