MTH5P1: Polynomials

Polynomials

Polynomials appear throughout Mathematics. This lesson covers some of the tools that will be useful in later work.

Introduction

A polynomial is an expression which:

(where x a is variable, k is a constant and n a positive integer)

Every polynomial in one variable (eg ‘x’) is equivalent to a polynomial with the form:

Polynomials are often described by their degree of order.

This is the highest index of the variable in the expression.

(eg: containing x⁵ order 5, containing x⁷ order 7 etc.)

These are NOT polynomials:

3x²+x^1/2+x

The second term has an index which is not an integer(whole number).

5x^-2+2x^-3+x^-5

Indices of the variable contain integers which are not positive.

examples of polynomials:

x⁵+5x²+2x+3

(x⁷+4x²)(3x-2)

x+2x²–5x³+x⁴-2x⁵+7x⁶

Algebraic Long Division

If, f(x) the numerator and d(x) the denominator are polynomials

and the degree of d(x) <= the degree of f(x)

and d(x) does not = 0

then, two unique polynomials q(x) the quotient and r(x) the remainder exist.

So that:

Note – The degree of r(x) < the degree of d(x).

We say that d(x) divides evenly into f(x), when r(x)=0.

Example

The Remainder Theorem

If a polynomial f(x) is divided by (x-a), the remainder is f(a).

Example

Find the remainder when (2x³+3x+x) is divided by (x+4).

The reader may wish to verify this answer by using algebraic division.

The Factor Theorem (a special case of the Remainder Theorem)

(x−a) is a factor of the polynomial f(x) if f(a) = 0

Example

Example (a) would usually be referred to as a linear expression (inx), whilst (b) and (c) are quadratic expressions (or just ‘quadratics’).Example (d) would be described as a cubic, whilst (e) could be described as a 5th order (or 5th degree) polynomial (although the terms ‘quartic’ and ‘quintic’ are sometimes used for 4th and 5th order
polynomials, respectively).

If there is a term which is a number only (i.e. does not involve x),then this is referred to as the constant term. Thus the constant term in (d) is 1.
It is possible to have a polynomial involving some other letter, but x is by far the most commonly used.
Polynomials can be added or subtracted, by grouping together the same powers of x.

The Factor Theorem

The Remainder Theorem

Then we know that g(x) = (x – 1)(x + 2)(x – 7) + 1
The remainder when g(x) is divided by x – 1, x + 2 or x – 7 is 1.
And g(1) = 0 + 1 = 1, since the factor x – 1 becomes 0 when x = 1.Similarly, g(–2) and g(7) also equal 1.
In general, if the polynomial g(x) is divided by (x – a), then the remainder will be g(a).

This is the Remainder Theorem.The Factor Theorem is in fact the special case of the Remainder
Theorem where the remainder is zero.
Example 10

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