Sign In | Starter Of The Day | Tablesmaster | Fun Maths | Maths Map | Topics | More

International Baccalaureate Mathematics

Functions

Syllabus Content

Polynomial functions, their graphs and equations; zeros, roots and factors. The factor and remainder theorems. Sum and product of the roots of polynomial equations.

Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.

Here is an exam-style questions on this statement:

Click on a topic below for suggested lesson Starters, resources and activities from Transum.


Furthermore

Official Guidance, clarification and syllabus links:

For the polynomial equation:

$$ \sum_{r=0}^n a_rx^r=0 $$

the sum is \( \frac{-a_{n-1}}{a_n} \)

the product is \( \frac{(-1)^na_0}{a_n} \)

Link to: complex roots of quadratic and polynomial equations (AHL 1.14).

Formula Booklet:

Sum and product of the roots of polynomial equations of the form

$$ \sum_{r=0}^n a_rx^r=0 $$

Sum is \( \frac{-a_{n-1}}{a_n} \); product is \( \frac{(-1)^na_0}{a_n} \)

The factor theorem and remainder theorem are fundamental concepts in polynomial algebra. The factor theorem states that for a polynomial \( P(x) \), if \( P(a) = 0 \), then \( (x-a) \) is a factor of \( P(x) \). The remainder theorem, on the other hand, states that when a polynomial \( P(x) \) is divided by a linear divisor \( x-a \), the remainder is \( P(a) \).

Key Formulae:

$$ P(a) = 0 \implies (x-a) \text{ is a factor of } P(x) $$ $$ \text{Remainder of } P(x) \text{ when divided by } (x-a) = P(a) $$

Example:

Consider the polynomial \( P(x) = x^2 - 3x + 2 \). Using the factor theorem, if \( P(1) = 0 \), then \( (x-1) \) is a factor of \( P(x) \).

$$ P(1) = 1^2 - 3(1) + 2 = 0 $$

Thus, \( (x-1) \) is a factor of \( P(x) \). Additionally, using the remainder theorem, the remainder when \( P(x) \) is divided by \( x-1 \) is \( P(1) \), which is 0 in this case.

How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers anywhere in the world. Click here to enter your comments.


Apple

©1997-2024 WWW.TRANSUM.ORG