The extended distributive law

The extended distributive law multiplies two sums: every term of the first by every term of the second.

\displaystyle (a+b)(c+d)

\displaystyle =

\displaystyle ac+ad+bc+bd

It is the distributive law $\displaystyle {a(b+c)}={ab+ac}$ , used twice.

The extended distributive law in areas

A rectangle is $\displaystyle x+4$ wide and $\displaystyle x+2$ high. Its area is $\displaystyle (x+4)(x+2)$ . But the rectangle is made of four rectangles, with the areas $\displaystyle x^2$ , $\displaystyle 4x$ , $\displaystyle 2x$ and $\displaystyle 8$ . So $\displaystyle {(x+4)(x+2)}={x^2+4x+2x+8}={x^2+6x+8}$ .

Every term by every term

Two terms times two terms make four products.

\displaystyle (x+4)(x+2)

is not

\displaystyle x^2+8

Open the brackets in two steps, then collect like terms:

The first bracket’s terms, each times the whole second bracket: $\displaystyle {(2x+3)(x+5)}={2x(x+5)+3(x+5)}$ .
The distributive law in each: $\displaystyle 2x^2+10x+3x+15$ .
Like terms collected: $\displaystyle 2x^2+13x+15$ .

With practice you write the four products straight away.

Exercise 1

Open the brackets and collect like terms.

/ – 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

Brackets with subtraction and negative numbers

Minus times minus is plus:

\displaystyle {(-4)\cdot(-2)}={+8}

Every term is multiplied with its sign:

$\displaystyle {(x-3)(x+5)}={x^2+5x-3x-15}={x^2+2x-15}$
$\displaystyle {(x-4)(x-2)}={x^2-2x-4x+8}={x^2-6x+8}$

Exercise 2

Open the brackets and collect like terms.

/ – 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

Brackets with three terms

The same law holds for brackets with more terms: two terms times three terms make six products.

(x+2)(x^2+3x+1)=x^3+3x^2+x+2x^2+6x+2=x^3+5x^2+7x+2

A product of three brackets: multiply two of them first, then the result by the third.

Exercise 3

Open the brackets and collect like terms.

/ – 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

Simplifying expressions

A minus before a product: open the product inside brackets first, then flip every sign:

\displaystyle {-(x+1)(x+2)}={-(x^2+3x+2)}={-x^2-3x-2}

To simplify an expression with products of brackets, open every product, then collect like terms:

(x+3)(x-1)-x(x+2)=x^2+2x-3-(x^2+2x)=x^2+2x-3-x^2-2x=-3

Exercise 4

Simplify.

/ – 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 5

A rectangle is $\displaystyle 2x+1$ long and $\displaystyle x+3$ wide.
Draw it cut into four parts, like the rectangle above, and write the area of each part.
Then write the rectangle’s area as an expression without 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

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