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Aims: To be able to solve partial fractions with repeated factors To be able to spot and cancel down in improper fraction before splitting it into it’s partial fractions To be able to solve binomial expansions with negative and fractional powers To be able to state the value of x for which the expansion will converge Algebra - Improper Factions & Binomial Expansion Lesson 3

Improper fractions Remember, an algebraic fraction is called an i_______ fraction if the degree of the polynomial is equal to, or greater than, the degree of the denominator. To express an improper fraction in partial fractions we start by expressing it in the algebraic equivalent of mixed number form. Express in partial fractions. Using long division:

Improper fractions Therefore

On w/b 1. Ex 2D pg 28 qu 4 parts a, c,d –6 minutes

Binomial expansion Previously in C2 we found that, when n is a positive whole number, If n is negative or fractional then, provided that | x | < 1, then it is an infinite series and will converge towards (1 + x ) n. This is a finite series with n + 1 terms.

Binomial expansion In general, for negative and fractional n and | x | < 1, Start by writing this as : Expand up to the term in x 4. This is equal to (1 + x ) –1 provided that | x | < 1. The expansion is then:

Binomial expansion Expand up to the term in x 3. Start by writing this as This converges towards provided that That is when | x | <.

On w/b Expand up to the term in x 3. This converges providingthat | x | <.

Binomial expansion Find the first four terms in the expansion of (3 – x ) –2. When the first term in the bracket is not 1, we have to factorize it first. For example:

Binomial expansion Therefore This expansion is valid for In general, if we are asked to expand an expression of the form ( a + bx ) n where n is negative or fractional we should start by writing this as: The corresponding binomial expansion will be valid for | x | <.

Binomial expansion Expand up to the term in x 2 giving the range of values for which the expansion is valid. This expansion is valid for

On w/b Expand up to the term in x 3. | x | <.This expansion is valid for 1.Complete valid limits puzzle 2.Do Ex 4B, p51