Algebraic Fractions Flashcards

All 22 cards in this deck

What is the first step in simplifying an algebraic fraction?

e.g. $\frac{x^2-x-6}{2x^2-5x-3}$

Factorise the numerator and the denominator fully.

e.g. $\frac{(x-3)(x+2)}{(2x+1)(x-3)}$

When are you allowed to cancel something from an algebraic fraction?

Only when it is a factor of the whole numerator and the whole denominator.

What are the steps to simplify an algebraic fraction?

e.g. $\frac{x^2-9}{x^2+3x}$

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

True or false? $\frac{x+2}{x}$ simplifies to $2$.

False. $x$ is not a factor of the whole numerator, so it cannot be cancelled.

True or false? $\frac{x^2-16}{x-4}$ simplifies to $x+4$.

True. $x^2-16=(x-4)(x+4)$, so the factor $(x-4)$ cancels.

How do you simplify a fraction of powers of the same letters?

e.g. $\frac{c^3d^4}{cd}$

Subtract the indices for each letter.

e.g. $c^{3-1}d^{4-1}=c^2d^3$

What are the steps to write a sum of algebraic fractions as a single fraction?

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

1. Common denominator: $x(x+1)$
2. Numerator: $2(x+1)+3x$
3. Expand and collect: $\frac{5x+2}{x(x+1)}$

How do you combine an integer with an algebraic fraction into a single fraction?

e.g. $6+\frac{x-1}{x-2}$

Write the integer over the same denominator, then add the numerators.

e.g. $\frac{6(x-2)+(x-1)}{x-2}=\frac{7x-13}{x-2}$

True or false? $\frac{3}{x+1}+\frac{1}{x-2}=\frac{4}{2x-1}$

False. You cannot add numerators and denominators; you must use a common denominator, here $(x+1)(x-2)$.

True or false? When subtracting algebraic fractions, the second numerator must be put in brackets before expanding.

True. The subtraction applies to every term of that numerator, e.g. $-(x^2+5x+6)=-x^2-5x-6$.

How do you add algebraic fractions with numerical denominators?

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

Use the lowest common multiple of the numbers as the denominator.

e.g. $\frac{5x+2(x+1)}{10}=\frac{7x+2}{10}$

What is the rule for multiplying two algebraic fractions?

e.g. $\frac{2}{x}\times\frac{x+1}{3}$

Multiply the numerators and multiply the denominators.

e.g. $\frac{2(x+1)}{3x}$

What is the rule for dividing by an algebraic fraction?

e.g. $\frac{x}{2}\div\frac{x+1}{3}$

Multiply by the reciprocal of the divisor (invert the second fraction and multiply).

e.g. $\frac{x}{2}\times\frac{3}{x+1}=\frac{3x}{2(x+1)}$

True or false? To divide one algebraic fraction by another, you invert the first fraction and multiply.

False. You invert the second fraction (the divisor) and multiply.

True or false? You need a common denominator before multiplying two algebraic fractions.

False. A common denominator is only needed for adding or subtracting.

What are the steps to multiply algebraic fractions and simplify?

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

1. Factorise: $\frac{(x-2)(x+2)}{x^2}\times\frac{3x}{x+2}$
2. Multiply numerators and denominators: $\frac{3x(x-2)(x+2)}{x^2(x+2)}$
3. Cancel common factors: $\frac{3(x-2)}{x}$

How do you divide an expression that is not a fraction by an algebraic fraction?

e.g. $(x+5)\div\frac{x^2+3x-10}{x-1}$

Write it over $1$, then multiply by the reciprocal.

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

What is the first step in solving an equation containing algebraic fractions?

Multiply every term by the common denominator to clear the fractions.

What are the steps to solve an equation containing algebraic fractions?

e.g. $\frac{2}{x}+\frac{3}{x+1}=2$

1. Multiply every term by $x(x+1)$: $2(x+1)+3x=2x(x+1)$
2. Expand and rearrange to $=0$: $2x^2-3x-2=0$
3. Solve the quadratic: $x=2$ or $x=-\frac{1}{2}$

What does cross-multiplying give for the equation $\frac{a}{b}=\frac{c}{d}$?

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

$ad=bc$

e.g. $2(x-2)=3(x+1)$

True or false? A value of $x$ that makes one of the denominators zero can still be a solution of the equation.

False. It must be rejected, because the fraction would be undefined.

True or false? Clearing the fractions in an equation with algebraic fractions always leaves a linear equation.

False. It often leaves a quadratic equation, which may have two solutions.