Algebraic fractions

Algebra, grade 9 Lesson 13 of 26Learning

Algebraic fractions

A fraction with a variable in its denominator is an algebraic fraction: $\displaystyle \frac{3}{x}$ , $\displaystyle \frac{x+1}{x-2}$ , $\displaystyle \frac{x^{2}-9}{x^{2}+3x}$ . Like a number fraction, it is a division: the numerator divided by the denominator. For every number we substitute for $\displaystyle x$ it becomes an ordinary fraction – for $\displaystyle {x}={4}$ , the fraction $\displaystyle \frac{x+1}{x-2}$ is $\displaystyle \frac{5}{2}$ .

The domain

Only the denominator matters. The numerator may be 0: for

\displaystyle {x}={-1}

the fraction

\displaystyle \frac{x+1}{x-2}

\displaystyle {\frac{0}{-3}}={0}

Division by 0 is not defined, so we may not substitute a number that makes the denominator 0. The numbers we may substitute are the domain of the fraction. To find it, find when the denominator is 0 – and leave those numbers out:

$\displaystyle \frac{5}{x-3}$ : the denominator is 0 when $\displaystyle {x}={3}$ . The domain: $\displaystyle x\neq3$ .
$\displaystyle \frac{x+1}{x^{2}-4}$ : factor the denominator, $\displaystyle {x^{2}-4}={(x-2)(x+2)}$ ; it is 0 when $\displaystyle {x}={2}$ or $\displaystyle {x}={-2}$ . The domain: $\displaystyle x\neq2$ , $\displaystyle x\neq-2$ .
$\displaystyle \frac{7}{x^{2}+5x}$ : $\displaystyle {x^{2}+5x}={x(x+5)}$ . The domain: $\displaystyle x\neq0$ , $\displaystyle x\neq-5$ .

Exercise 1

Find the domain of each fraction.
Factor the denominator where needed.

$\displaystyle \frac{4}{x+6}$

$\displaystyle \frac{x-1}{2x-8}$

$\displaystyle \frac{x}{x^{2}-25}$

$\displaystyle \frac{3}{x^{2}-6x}$

$\displaystyle \frac{x+2}{x^{2}+x-6}$

Reducing an algebraic fraction

A number fraction is reduced by dividing the numerator and the denominator by a common factor: $\displaystyle {\frac{6}{8}}={\frac{2\cdot3}{2\cdot4}}={\frac{3}{4}}$ . An algebraic fraction is reduced the same way. When the numerator and the denominator are products, you can see the common factor at once:

\frac{6x}{9x^{2}}=\frac{3x\cdot2}{3x\cdot3x}=\frac{2}{3x}\qquad(x\neq0)

Reducing a fraction with a sum in it

When the numerator or the denominator is a sum, first factor each of them, and only then cancel:

\frac{3x+6}{x^{2}+2x}=\frac{3(x+2)}{x(x+2)}=\frac{3}{x}\qquad(x\neq0,\ x\neq-2)

The domain is found from the original fraction, before reducing: $\displaystyle -2$ is left out even though the reduced fraction $\displaystyle \frac{3}{x}$ could take it.

Factors and terms in cancelling

Careful: cross out only factors, never a term of a sum.

\displaystyle \frac{x+3}{x}

is not

\displaystyle 3

You may cancel only a factor – something that multiplies the whole numerator and the whole denominator. A term of a sum may not be cancelled. Substitution shows it: for $\displaystyle {x}={1}$ , $\displaystyle {\frac{x+3}{x}}={\frac{4}{1}}={4}$ , not 3. The fraction $\displaystyle \frac{x+3}{x}$ cannot be reduced at all, and neither can $\displaystyle \frac{x+5}{x+2}$ .

Reducing: the steps

Factor the numerator and the denominator.
Find the domain: when is the denominator 0?
Cancel the common factors.

Exercise 2

Reduce the fractions: first write the numerator and the denominator as products, then cancel.
Write the domain of each fraction too.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Noa reduced like this: $\displaystyle {\frac{x+6}{6}}={x}$ .
Substitute $\displaystyle {x}={6}$ in both and show that she was wrong.
Can the fraction $\displaystyle \frac{x+6}{6}$ be reduced at all?

Formulas: algebra, grade 9

Open as a page

The laws of powers

Multiplying powers with the same base
$\displaystyle a^{m}\cdot a^{n}$ $\displaystyle =$ $\displaystyle a^{m+n}$
Dividing powers with the same base
$\displaystyle a^{m}:a^{n}$ $\displaystyle =$ $\displaystyle a^{m-n}$ $\displaystyle a\neq0$
A power of a power
$\displaystyle \left(a^{m}\right)^{n}$ $\displaystyle =$ $\displaystyle a^{m\cdot n}$
A power of a product
$\displaystyle (a\cdot b)^{n}$ $\displaystyle =$ $\displaystyle a^{n}\cdot b^{n}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{n}$ $\displaystyle =$ $\displaystyle \frac{a^{n}}{b^{n}}$ $\displaystyle b\neq0$
The power zero
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$

The laws of square roots

The root of a product
$\displaystyle \sqrt{a\cdot b}$ $\displaystyle =$ $\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle a\ge0,\ b\ge0$
The root of a quotient
$\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle =$ $\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle a\ge0,\ b>0$
The square of a root
$\displaystyle \left(\sqrt{a}\right)^{2}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$
The root of a square
$\displaystyle \sqrt{a^{2}}$ $\displaystyle =$ $\displaystyle a$ $\displaystyle a\ge0$

The extended distributive law and the short multiplication formulas

The extended distributive law
$\displaystyle (a+b)(c+d)$ $\displaystyle =$ $\displaystyle ac+ad+bc+bd$
The square of a sum
$\displaystyle (a+b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}+2ab+b^{2}$
The square of a difference
$\displaystyle (a-b)^{2}$ $\displaystyle =$ $\displaystyle a^{2}-2ab+b^{2}$
The difference of squares
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^{2}-b^{2}$

The parabola

Standard form
$\displaystyle y=ax^{2}+bx+c$
Vertex form – the vertex is $\displaystyle (b,k)$
$\displaystyle y=a(x-b)^{2}+k$
Product form – it meets the $\displaystyle x$ -axis at $\displaystyle x_1$ and $\displaystyle x_2$
$\displaystyle y=a(x-x_1)(x-x_2)$
The vertex’s $\displaystyle x$ , in standard form
$\displaystyle x=-\frac{b}{2a}$

The quadratic equation

The roots of $\displaystyle {ax^{2}+bx+c}={0}$ , when $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$