IB Mathematics AA HL Polynomials Study Notes| New Syllabus

IB Mathematics AA HL Polynomials Study Notes

IB Mathematics AA HL Polynomials Study Notes Offer a clear explanation of Polynomials , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Polynomials.

Polynomial Functions

A polynomial function is a function of the form:

$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $

where:

$ a_n, a_{n-1}, \ldots, a_1, a_0 $ are real coefficients
$ a_n \neq 0 $
$ n $ is a non-negative integer (degree of the polynomial)

Key Characteristics of Polynomial Functions

Degree (n): The highest power of $ x $; determines the shape and end behavior of the graph.
Leading coefficient (aₙ): The coefficient of the term with the highest degree; affects the graph’s orientation and steepness.
Constant term (a₀): The y-intercept, where the graph crosses the y-axis at $ (0, a_0) $.
Continuous & smooth: Polynomial graphs have no breaks, holes, or sharp corners.

Zeros, Roots and Factors

Zero/root: A value $ x = r $ where $ f(r) = 0 $. The graph crosses or touches the x-axis at these points.
Factor: If $ r $ is a root, then $ (x – r) $ is a factor of the polynomial.
The Fundamental Theorem of Algebra states that a polynomial of degree $ n $ has exactly $ n $ complex roots (some may be repeated).

End Behavior

The end behavior of the graph depends on the degree and leading coefficient:

If $ n $ is even and $ a_n > 0 $: both ends rise ($ y \to +\infty $)
If $ n $ is even and $ a_n < 0 $: both ends fall ($ y \to -\infty $)
If $ n $ is odd and $ a_n > 0 $: left falls, right rises
If $ n $ is odd and $ a_n < 0 $: left rises, right falls

Multiplicity of Roots

Multiplicity 1 (simple root): Graph crosses x-axis at that root.
Multiplicity even: Graph touches x-axis at that root and turns back.
Multiplicity odd (≥ 3): Graph crosses but flattens near the root.

Example

Consider the polynomial function:

$ f(x) = (x – 1)(x + 2)^2(x – 3) $ Find Degrees , Roots and Behaviour

▶️Answer/Explanation

Degree: 4 (sum of multiplicities: 1 + 2 + 1)
Roots:
- $ x = 1 $ (simple root, graph crosses the x-axis)
- $ x = -2 $ (multiplicity 2, graph touches the x-axis and turns)
- $ x = 3 $ (simple root, graph crosses the x-axis)
Y-intercept: $ f(0) = (0 – 1)(0 + 2)^2(0 – 3) = (-1)(4)(-3) = 12 $ → Point: (0, 12)
End behavior: Degree is even and leading coefficient is positive → both ends rise as $ x \to \pm \infty $.

The graph:

Crosses the x-axis at $ x = 1 $ and $ x = 3 $.
Touches and turns at $ x = -2 $ because of multiplicity 2.
Passes through the y-intercept at (0, 12).
Rises to infinity on both ends.

Factor and Remainder theorems.

Factor Theorem: If a polynomial function $ f(x) $ has $ f(a) = 0 $, then $ (x – a) $ is a factor of $ f(x) $.

Remainder Theorem: When a polynomial $ f(x) $ is divided by $ (x – a) $, the remainder is $ f(a) $.

How they are connected:

If $ f(a) = 0 $, remainder is 0 → $ (x – a) $ is a factor → Factor Theorem applies.
If $ f(a) \neq 0 $, $ f(a) $ is the remainder when divided by $ (x – a) $.

Applications:

Quickly test if a linear binomial is a factor.
Efficiently compute remainders without full division.
Assist in factorization of higher-degree polynomials.

Example

Given the polynomial $ f(x) = 2x^3 – 3x^2 – 2x + 3 $, use the Factor and Remainder Theorems to check whether $ (x – 1) $ and $ (x + 3) $ are factors of $ f(x) $.

▶️Answer/Explanation

Test $ (x – 1) $

Compute $ f(1) $:

$ f(1) = 2(1)^3 – 3(1)^2 – 2(1) + 3 = 2 – 3 – 2 + 3 = 0 $

Since $ f(1) = 0 $, $ (x – 1) $ is a factor.

Test $ (x + 3) $

Compute $ f(-3) $:

$ f(-3) = 2(-3)^3 – 3(-3)^2 – 2(-3) + 3 = 2(-27) – 3(9) + 6 + 3 = -54 – 27 + 6 + 3 = -72$

Since $ f(-3) = -72 $, $ (x + 3) $ is not a factor. The remainder is -72.

Sum and Product of the Roots of Polynomial Equations.

Consider a polynomial equation of degree $ n $:

$a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_1 x + a_0 = 0 $

Let the roots be $ \alpha_1, \alpha_2, \dots, \alpha_n $.

Sum of the roots: $ \alpha_1 + \alpha_2 + \cdots + \alpha_n = -\frac{a_{n-1}}{a_n} $
Sum of products of roots two at a time: $ \sum \alpha_i \alpha_j = \frac{a_{n-2}}{a_n} $ (where the sum is over all pairs)
Sum of products of roots three at a time: $ \sum \alpha_i \alpha_j \alpha_k = -\frac{a_{n-3}}{a_n} $ (sum over all triplets)
Product of all roots: $ (-1)^n \frac{a_0}{a_n} $

These are known as Vieta’s formulas.

Some Specefic Polynomial Functions:

Quadratic: $ ax^2 + bx + c = 0 $

Sum: $ -\frac{b}{a} $
Product: $ \frac{c}{a} $

Cubic: $ ax^3 + bx^2 + cx + d = 0 $

Sum: $ -\frac{b}{a} $
Sum of products two at a time: $ \frac{c}{a} $
Product: $ -\frac{d}{a} $

Quartic: $ ax^4 + bx^3 + cx^2 + dx + e = 0 $

Sum: $ -\frac{b}{a} $
Sum of products two at a time: $ \frac{c}{a} $
Sum of products three at a time: $ -\frac{d}{a} $
Product: $ \frac{e}{a} $

Example:

Given the quadratic equation:

$ 2x^2 – 5x + 3 = 0 $

Find the sum and product of its roots.

▶️Answer/Explanation

Here, $ a = 2 $, $ b = -5 $, $ c = 3 $.

Sum of roots: $ -\frac{b}{a} = -\frac{-5}{2} = \frac{5}{2} $
Product of roots: $ \frac{c}{a} = \frac{3}{2} $

Example:

Given the cubic equation:

$ x^3 – 6x^2 + 11x – 6 = 0 $

Find the sum, sum of products of roots two at a time, and product of roots.

▶️Answer/Explanation

Here, $ a = 1 $, $ b = -6 $, $ c = 11 $, $ d = -6 $.

Sum of roots: $ -\frac{b}{a} = -\frac{-6}{1} = 6 $
Sum of products of roots two at a time: $ \frac{c}{a} = \frac{11}{1} = 11 $
Product of roots: $ -\frac{d}{a} = -\frac{-6}{1} = 6 $

IB Mathematics AA HL Polynomials Study Notes| New Syllabus