Aims:• To recap the remainder and factor theorems.• To be able to divide a polynomial by another polynomialusing your preferred method• To understand terminology, quotient and remainder.

Algebra - Long Division Lesson 2

Improper fractions and mixed numbersRemember, a numerical fraction is called an improper fraction if the numerator is larger than the denominator and we can change them to a mixed number.

For example, 296

Now ___ is the quotient and ___ is the remainder.

f rqg g( ) ( )( )+( ) ( )x xxx x

An algebraic fraction is called an improper fraction when the numerator is a polynomial of degree greater than, or equal to, the degree of the denominator and in general is written as:

Write in the form A + .+ 3

1xx 1

Bx

Rewriting the numeratorA useful technique for writing improper fractions in proper form is to look for ways to rewrite the numerator so that part of it can be divided by the denominator. For example,

+ 31

xx

Rewriting the numerator

We can write this in fraction form as:2

2

3 + 2+1

x xx

So when 3x2 + 2x is divided by x2 + 1 the quotient is ___ and the remainder is _________.

What is the quotient and the remainder when 3x2 + 2x is divided by x2 + 1?

Constructing an identityWhen the numerator cannot easily be manipulated we can use long division. For example

What is x3 – 4x2 + 5 divided by x2 – 3?

x3 – 4x2 + 0x + 5x2 – 3

The quotient is and the remainder is so, 3

2 2

2 3 74 +3

4 + 53

x xx

xxx

What is x4 + 2x3 – 4x2 + 5 divided by x2 – 4x ?

Long Division

What is x4 + 3x3 – 2x + 6 divided by x2 + 2x ?

On w/bDo Ex B page 21