International Baccalaureate Mathematics

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.

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Polynomial Division Practise dividing one algebraic expression by another in this set of exercises.

Here is an exam-style questions on this statement:

"Consider the equation $ (k-4)x^2 — \frac{kx}{5} + 4(2-k) = 0 $, where $ k \in \mathbb{R} $. " ... more

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Algebra Pupils begin their study of algebra by investigating number patterns. Later they construct and express in symbolic form and use simple formulae involving one or many operations. They use brackets, indices and other constructs to apply algebra to real word problems. This leads to using algebra as an invaluable tool for solving problems, modelling situations and investigating ideas. If this topic were split into four sub topics they might be: Creating and simplifying expressions; Expanding and factorising expressions; Substituting and using formulae; Solving equations and real life problems; This is a powerful topic and has strong links to other branches of mathematics such as number, geometry and statistics. See also "Number Patterns", "Negative Numbers" and "Simultaneous Equations".

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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.

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International Baccalaureate Mathematics

Functions

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