Mathematics: Core (Notes 1 of 1)

To simplify algebraic fractions:-Factorise-Look for common factors in the top and bottom and cancel-To divide by a fraction multiply by the reciprocal-To add and subtract fractions find a common denominatorTo divide polynomials use algebraic long division or compare coefficients (algebraic juggling is also an option) but read the question to see if a certain method is specified. A polynomial can be written as (Quotient)(Divisor) + Remainder. The divisor is the thing you divide by, the quotient is the answer, and the remainder is the bit left over.One way of 'simplifying' algebraic fractions is by expressing them as partial fractions. To do this set the original fraction equal to 2 fractions with the denominator split, write as two fractions with numerators A and B, multiply both sides up to get rid of the fractions and then let x=a number to make one of the brackets 0 OR compare coefficients. Parametric equations are when x and y are defined separately (in two equations) in terms of another parameter (usually t or $\theta$. To find where graphs intersect equate the two equations, to find points of intersection with the axis set the equations equal to 0 and work out the t value then substitute it into the other equation. Rearrange a parametric equation to make t the subject then substitute it into the other equation (using trig identities if required).The binomial expansion formula is given in the formula book for the bracket:

$$(1+x)^n$$ To use this the number at the front MUST be 1, if it isn't factorise the number out of the bracket to make it 1, but remember that when you factorise out the number is to the power n as well as the bracket left over.

To integrate implicitly wrt x: You can differentiate every term on both side of the equals sign without having to rearrange to get y=an expression. Differentiate $x^n$ terms as normal.Use the CHAIN RULE to differentiate $y^n$ terms, ie. 'differentiate wrt y and then write $\frac{dy}{dx}$Use the PRODUCT RULE to differentiate terms with both x and y included.To differentiate parametric equations wrt x: You don't have to find the Cartesian equation first (which might be difficult to differentiate anyway)Use the CHAIN RULE: $\frac{dy}{dx}$ = $\frac{dy}{dt}$ x $\frac{dt}{dx}$Remember that $\frac{dt}{dx}$ is just $\frac{1}{\frac{dx}{dt}}$Differentiate both parametric equations separately and then multiply the results.Differentiating trig equations (you just have to learn the results for sin, cos and tan):The results for cosec, sec, and cot are in the formula book but can be found using PRODUCT RULE.The reason it is important to be able to express a function as partial fractions is because even if the original fraction can't be integrated the two (or more) fractions which it equals can be integrated.  

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Algebra and Graphs

Differentiation and Integration