# 1.2 Polynomials

A polynomial is an expression with

- only one variable (eg x)
- one or more terms
- each term has a non-negative, integer power of x

In the polynomial, P(x)

- a
_{n}is the coefficient of x^{n}, etc - P(x) has a degree of n (the highest power of x)

Polynomials are usually named with consecutive capital letters, starting at P (ie P, Q, R, etc)

You are already familiar with many polynomials:

- a polynomial with degree 1 is Linear
- a polynomial with degree 2 is a Quadratic
- a polynomial with degree 3 is a Cubic
- a polynomial with degree 4 is a Quartic

## Factorising Quadratics

Recall that we can factorise quadratics using the following techniques:

### 0. Common Factors

When factorising any polynomial, always take out any common factors first.

### 1. Perfect Squares

(2)Example:

(3)### 2. Difference of Perfect Squares

(4)Example:

(5)### 3. Shortcut Approach

For a full review of this approach when a = 1 go here

Example:

(6)For a full review of this approach when a <> 1, go here

Example:

(8)### 4. Completing the Square

- This method will factorise any quadratic that can be factorised but use a quicker method if you can
- Note that it assumes a = 1, so if a <> 1 take that out as a common factor first
- For a full review of this approach, go here.

Example:

(10)- Note that if you end up with a plus sign between the two terms in the 2nd last step,
- then the quadratic can not be factorised (in the Real number system)

## Factorising Cubics

We can factorise cubic polynomials using the following techniques

### 1. Perfect Cubes

(11)Example

(12)### 2. Sum and Difference of Two Cubes

(13)Example

(14)### 3. Factorising Cubics by Grouping 2 and 2

- only some cubics can be factorised in this way

Example

(15)## Factor Theorem

If factorising a higher degree polynomial, P(x), or the previous methods aren't effective, we can apply the factor theorem.

- If P(a) = 0 then the polynomial P(x) has a factor of (x – a)

NOTE: if (x – a) is a factor of P(x) then a must be a factor of the constant term in P(x).

To find a factor of P(x) use trial and error with different factors (positive and negative) of the constant term in P(x)

Example

(16)The constant term is 6, factors of 6 are: 1, –1, 2, –2, 3, –3, 6, –6

Start with the easiest values first

## Shortcut Method for Factorising Cubics when a factor is known

Once the factor theorem has revealed one factor, we can complete the factorisation using the following method

For a demonstration of this process in Powerpoint, download the following file (2.4 Mb)

Example

- Factorise $(x^3 – 2x^2 – 3x + 6)$ given that $(x – 2)$ is a factor

Solution:

By examination of the above, we can conclude that the other factor must be a quadratic.

Further the first term must be x^{2} and the last term must be –3 so we can write:

$x^3 – 2x^2 – 3x + 6 = (x – 2)(x^2 + ax – 3)$ where a is an unknown

If we expand these brackets we get:

(19)Equate the coefficients of x^{2} (underlined above), we get:

–2 = a – 2

so a = 0

Thus

(20)The quadratic in the second bracket can then be factorised

in this case using difference of two squares

Thus

(21)## Solving Polynomial Equations

Once any type of polynomial equation has been factorised, it can be solved using the Null Factor Law.

Example

(22)## The Quadratic Formula

For any quadratic equation in the form:

(23)We can use the Quadratic Formula to solve it:

(24)Recall that the Discriminant gives us information about the number and type of solutions:

(25)- if discriminant less than zero, there are no real solutions
- if discriminant equal to zero, there is one real solution
- the solution is rational
- if discriminant greater than zero, there are two real solutions
- if discriminant is a perfect square, the solutions are rational
- if discriminant is not a perfect square, the solutions are irrational

## Equivalent Polynomials

Two polynomials are equivalent if and only if each of the respective coefficients are equal.

The notation to say that P(x) is equivalent to Q(x) is: $\; P(x) \equiv Q(x)$

Example

Find the value of k if:$3x^2 + kx - 5 \equiv 3x^2 - 4x - 5$

Solution: The equivalent coefficient of x is –4, so k = –4.

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