When two taps fill a tank together, or two players complete a task together in a video game at different speeds, the time they take together is found by adding fractions in which the unknown appears in the denominator. That is an algebraic fraction: an ordinary fraction, but with polynomials on the top and bottom instead of just numbers.
To simplify an algebraic fraction, you factorise the numerator and the denominator and cancel the factors that appear on both the top and the bottom. It is exactly what you do with numerical fractions: $\dfrac{12}{18} = \dfrac{2\cdot 2\cdot 3}{2\cdot 3\cdot 3} = \dfrac{2}{3}$.
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Simplify the algebraic fraction $\frac{x^2 - 4}{x + 2}$.
Simplify the algebraic fraction $\frac{x^2 - 4}{x + 2}$.
A student claims that $\frac{x + 5}{5} = x + 1$, because "the $5$ on top and bottom cancel out". State whether her claim is true or false by checking it with a specific…
A student claims that $\frac{x + 5}{5} = x + 1$, because "the $5$ on top and bottom cancel out". State whether her claim is true or false by checking it with a specific value of $x$, explain the error, and write the correct simplification.
A fraction does not exist if its denominator is $0$. This places restrictions on the values that $x$ can take, and these restrictions are read from the original denominator, before simplifying.
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Check whether the algebraic fractions $\frac{x - 1}{x^2 - 1}$ and $\frac{1}{x + 1}$ are equivalent. Justify your answer.
Check whether the algebraic fractions $\frac{x - 1}{x^2 - 1}$ and $\frac{1}{x + 1}$ are equivalent. Justify your answer.
Just as with numbers: to multiply, multiply numerator by numerator and denominator by denominator; to divide, multiply by the reciprocal fraction. It is best to factorise before multiplying, so that you simplify along the way instead of at the end with bigger polynomials.
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Calculate and simplify $\frac{2x}{3} \cdot \frac{6}{x^2}$.
Calculate and simplify $\frac{2x}{3} \cdot \frac{6}{x^2}$.
Just as with numbers, you need a common denominator. If the denominators are not the same, you factorise them and find the lowest common multiple of the factors.
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Work out and simplify $\frac{2}{x} + \frac{3}{x + 1}$.
Work out and simplify $\frac{2}{x} + \frac{3}{x + 1}$.
Back to the start: If one player takes $x$ hours and the other takes $2$ hours to do a task, in one hour together they complete $\dfrac{1}{x} + \dfrac{1}{2} = \dfrac{x+2}{2x}$ of the total: a sum with the unknown in the denominator that is solved with the rules you have just learned.
Summary
Let's practice! 10 medium exercises in this section