Factoring: a common factor

Factoring an expression means writing it as a product. It is the distributive law read backwards:

opening brackets: $\displaystyle {3(x+4)}={3x+12}$ ;
factoring: $\displaystyle {3x+12}={3(x+4)}$ .

The product is equal to the original expression: opening its brackets gives the original back.

The greatest common factor of numbers

\displaystyle {12x+18}={2(6x+9)}

is equal too, but not complete:

\displaystyle 6x+9

still has the common factor 3. Take out the greatest.

The common factor is the greatest number that divides every term. In $\displaystyle 12x+18$ , 6 divides both 12 and 18:

$\displaystyle {12x+18}={6(2x+3)}$
in the brackets, each term divided by the factor: $\displaystyle {12x:6}={2x}$ , $\displaystyle {18:6}={3}$ .

A common factor with variables

A power is a product, $\displaystyle {x^{2}}={x\cdot x}$ , so a variable can be a common factor too:

$\displaystyle {x^{2}+5x}={x(x+5)}$ – $\displaystyle x$ is in both terms;
$\displaystyle {x^{3}-4x^{2}}={x^{2}(x-4)}$ – the common factor is the lowest power of $\displaystyle x$ .

Numbers and variables together

The common factor of the numbers and of the variables is taken out at once: $\displaystyle {6x^{2}+9x}={3x(2x+3)}$ – 3 divides 6 and 9, and $\displaystyle x$ is in both terms. The rectangle’s area is $\displaystyle 6x^{2}+9x$ : its height, $\displaystyle 3x$ , times its width, $\displaystyle 2x+3$ .

Checking a factorisation

Check that the factor you took out is the greatest one.

Open the brackets of the result: the original expression should come back. $\displaystyle {3x(2x+3)}={6x^{2}+9x}$ – right.

Exercise 1

Factor: take out the greatest common factor.

/ – 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

A term that is all common factor

Without the 1 the check fails:

\displaystyle {4\cdot x}={4x}

, not

\displaystyle 4x+4

When a whole term is the common factor, 1 is left in its place: $\displaystyle {4x+4}={4(x+1)}$ ; $\displaystyle {x^{2}-x}={x(x-1)}$ .

A common factor of three terms, and a negative one

With three terms, the factor divides all three: $\displaystyle {2x^{2}-6x+10}={2(x^{2}-3x+5)}$ .
A negative common factor flips the signs in the brackets: $\displaystyle {-2x-8}={-2(x+4)}$ .

Exercise 2

Factor, and check by opening the brackets.

/ – 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

Dan factored like this: $\displaystyle {6x^{2}+3x}={3x\cdot2x}$ .
Multiply out his product, and show that it is not equal to $\displaystyle 6x^{2}+3x$ .
Explain the mistake.
Factor correctly.

/ – 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

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 $\displaystyle a\neq0$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$