Factoring by the short multiplication formulas

Algebra, grade 9 Lesson 10 of 26Learning

Factoring by the short multiplication formulas

The short multiplication formulas, read from right to left, are factorisations:

\displaystyle (a+b)(a-b)

\displaystyle =

\displaystyle a^{2}-b^{2}

\displaystyle (a+b)^{2}

\displaystyle =

\displaystyle a^{2}+2ab+b^{2}

\displaystyle (a-b)^{2}

\displaystyle =

\displaystyle a^{2}-2ab+b^{2}

A difference of squares

A sum of squares, such as

\displaystyle x^{2}+25

, has no such factoring.

Two terms, each a square, with a minus between them, factor into a sum times a difference:

$\displaystyle {x^{2}-25}={x^{2}-5^{2}}={(x+5)(x-5)}$
$\displaystyle {4x^{2}-9}={(2x)^{2}-3^{2}}={(2x+3)(2x-3)}$
$\displaystyle {49-y^{2}}={(7+y)(7-y)}$

Exercise 1

Factor with the difference of squares.

/ – 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 perfect square trinomial

Check the middle term. In

\displaystyle x^{2}+10x+16

\displaystyle {16}={4^{2}}

, but

\displaystyle {2\cdot x\cdot4}={8x}

, not

\displaystyle 10x

– not a square.

A trinomial is a square of a bracket when its first and last terms are squares and its middle term is twice their product:

$\displaystyle x^{2}+10x+25$ : $\displaystyle x^{2}$ and $\displaystyle 5^{2}$ , and $\displaystyle {2\cdot x\cdot5}={10x}$ , so it is $\displaystyle (x+5)^{2}$ ;
$\displaystyle {x^{2}-12x+36}={(x-6)^{2}}$ – a minus in the middle gives a minus in the bracket;
$\displaystyle {9x^{2}+12x+4}={(3x+2)^{2}}$ – here the coefficient of $\displaystyle x^{2}$ is not 1, and the formula still works.

Exercise 2

Write each trinomial as a square of a bracket.

/ – 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

The order of factoring

Factoring is complete when no bracket can be factored further.

Take out a common factor, if there is one.
Look at what is left in the brackets:
- two terms – a difference of squares;
- three terms – a square of a bracket, or a trinomial by sum and product.

For example:

$\displaystyle {3x^{2}-12}={3(x^{2}-4)}={3(x+2)(x-2)}$
$\displaystyle {2x^{2}+12x+18}={2(x^{2}+6x+9)}={2(x+3)^{2}}$
$\displaystyle {x^{3}-x}={x(x^{2}-1)}={x(x+1)(x-1)}$

Exercise 3

Factor completely: a common factor first, then a formula.

/ – 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 4

Calculate in your head by factoring a difference of squares, for example $\displaystyle {21^{2}-19^{2}}={(21+19)(21-19)}={40\cdot2}={80}$ .

/ – 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

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