Algebraic Fractions Flashcards

All 7 cards in this deck

What are the steps to write an expression made of whole terms and fractions as a single fraction in simplest form?
e.g. $2 + \frac{3}{x} - \frac{1}{2x}$

1. Choose a common denominator: $2x$
2. Rewrite every term over it: $\frac{4x}{2x} + \frac{6}{2x} - \frac{1}{2x}$
3. Combine the numerators into one fraction: $\frac{4x+5}{2x}$

What are the steps to simplify fully a fraction with a quadratic numerator and a quadratic denominator?
e.g. $\frac{x^2+4x+3}{x^2-9}$

1. Factorise the numerator into two brackets: $(x+3)(x+1)$
2. Factorise the denominator into two brackets: $(x+3)(x-3)$
3. Cancel the common bracket: $\frac{x+1}{x-3}$

What are the steps to divide one algebraic fraction by another and simplify fully?
e.g. $\frac{4}{a+2} \div \frac{8}{a^2-4}$

1. Multiply by the reciprocal of the second fraction: $\frac{4}{a+2} \times \frac{a^2-4}{8}$
2. Factorise and write as one fraction: $\frac{4(a+2)(a-2)}{8(a+2)}$
3. Cancel common factors: $\frac{a-2}{2}$

True or false? $\frac{\sin x(3\cos x + \sin x)}{4\cos x(3\cos x + \sin x)}$ simplifies to $\frac{\tan x}{4}$.

True. The common bracket cancels to give $\frac{\sin x}{4\cos x}$, and $\frac{\sin x}{\cos x} = \tan x$.

What are the steps to subtract/add algebraic fractions where one denominator is quadratic and factorises?

e.g. $\frac{1}{x-2} + \frac{3}{x^2-4}$

1. Factorise the quadratic denominator: $x^2-4=(x-2)(x+2)$
2. Write both over the common denominator $(x-2)(x+2)$: $\frac{x+2}{(x-2)(x+2)}+\frac{3}{(x-2)(x+2)}$
3. Combine the numerators and simplify: $\frac{x+5}{(x-2)(x+2)}$

What are the steps to show an expression simplifies to a given form and to find the constant $k$?

e.g. show $\frac{2}{x-1}-\frac{2}{x+1}=\frac{k}{x^2-1}$

1. Write as a single fraction over the common denominator: $\frac{2(x+1)-2(x-1)}{(x-1)(x+1)}$
2. Expand and simplify the numerator: $\frac{4}{x^2-1}$
3. Compare with the given form: $k=4$

What are the steps to solve an equation where one algebraic fraction equals another?

e.g. $\frac{x+2}{5}=\frac{x-1}{2}$

1. Cross-multiply: $2(x+2)=5(x-1)$
2. Expand both brackets: $2x+4=5x-5$
3. Collect terms and solve: $x=3$