BeginnerTopic 10 of 104Engineering Mathematics
Polynomials and factoring
Breaking polynomials into factors reveals roots and simplifies rational expressions.
In depth
A polynomial is a sum of terms cₙxⁿ. Factoring writes it as a product of simpler polynomials, revealing roots. Techniques include common factors, difference of squares, grouping and the rational root theorem. A degree-n polynomial has n roots counting complex ones.
Key points
- Factor theorem: (x-r) divides p(x) if p(r)=0
- Synthetic division speeds up testing roots
- Characteristic equations of systems are polynomials
Examplex³-6x²+11x-6 = (x-1)(x-2)(x-3).
What is Polynomials and factoring?
Breaking polynomials into factors reveals roots and simplifies rational expressions.
Why is it important to understand polynomials and factoring?
Factoring exposes the roots of a polynomial, and roots are the poles and zeros that decide how a system behaves.
Why does factoring matter in control engineering?
The roots of the denominator are the system poles. Their locations determine whether a response is fast, slow, oscillatory or unstable.
Is there always a method to factor a polynomial by hand?
Not always. Higher-degree polynomials often need numerical root-finding, which is why software is used for them.
How is Polynomials and factoring used in practice?
Typical applications include finding the poles of a transfer function from its denominator, simplifying a rational expression before integrating and locating equilibrium points of a nonlinear model.
What should I learn before Polynomials and factoring?
Study "Exponents and logarithms" first (topic 9 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Polynomials and factoring?
Factor theorem: (x-r) divides p(x) if p(r)=0. Synthetic division speeds up testing roots. Characteristic equations of systems are polynomials.
Can you give an example of Polynomials and factoring?
x³-6x²+11x-6 = (x-1)(x-2)(x-3).
What comes after Polynomials and factoring?
The next topic is "Inequalities", which builds directly on this one.
How difficult is Polynomials and factoring?
It is classed as Beginner in Engineering Mathematics (topic 10 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.