Factoring a trinomial

A trinomial is an expression with three terms, such as $\displaystyle x^{2}+7x+12$ . A trinomial $\displaystyle x^{2}+bx+c$ is factored into two brackets, $\displaystyle (x+m)(x+n)$ . The extended distributive law shows where $\displaystyle m$ and $\displaystyle n$ come from:

(x+m)(x+n)=x^{2}+mx+nx+mn=x^{2}+(m+n)x+mn

So you look for two numbers with:

sum $\displaystyle {m+n}={b}$ , the coefficient of $\displaystyle x$ ;
product $\displaystyle {m\cdot n}={c}$ , the free term (the number with no $\displaystyle x$ ).

Finding the two numbers

Start from the product: it has fewer pairs than the sum.

In $\displaystyle x^{2}+7x+12$ , list the pairs whose product is 12, and pick the pair whose sum is 7:

Pair	1, 12	2, 6	3, 4
Sum	13	8	7

So $\displaystyle {x^{2}+7x+12}={(x+3)(x+4)}$ . Check: $\displaystyle {(x+3)(x+4)}={x^{2}+4x+3x+12}={x^{2}+7x+12}$ .

The signs of the two numbers

The signs of $\displaystyle b$ and $\displaystyle c$ give the signs of the two numbers:

$\displaystyle c$ positive, $\displaystyle b$ positive – both positive: $\displaystyle {x^{2}+7x+12}={(x+3)(x+4)}$ ;
$\displaystyle c$ positive, $\displaystyle b$ negative – both negative: $\displaystyle {x^{2}-8x+15}={(x-3)(x-5)}$ ;
$\displaystyle c$ negative – one positive and one negative, and the one further from 0 has the sign of $\displaystyle b$ : $\displaystyle {x^{2}+2x-15}={(x+5)(x-3)}$ , $\displaystyle {x^{2}-2x-15}={(x-5)(x+3)}$ .

Exercise 1

Factor the trinomials.

/ – 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 trinomial whose two numbers are equal

The two numbers can be equal. In $\displaystyle x^{2}+6x+9$ : $\displaystyle {3+3}={6}$ and $\displaystyle {3\cdot3}={9}$ , so $\displaystyle {x^{2}+6x+9}={(x+3)(x+3)}={(x+3)^{2}}$ – a square of a bracket.

A trinomial with a common factor

The sum-and-product method needs the coefficient of

\displaystyle x^{2}

to be 1.

When all the terms have a common factor, take it out first, then factor the trinomial in the brackets:

$\displaystyle {2x^{2}+10x+12}={2(x^{2}+5x+6)}={2(x+2)(x+3)}$
$\displaystyle {x^{3}-5x^{2}+6x}={x(x^{2}-5x+6)}={x(x-2)(x-3)}$

Exercise 2

Take out a common factor, then factor the trinomial.

/ – 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 trinomial with no whole-number pair

Some trinomials have no pair of whole numbers that fits. In $\displaystyle x^{2}+4x+5$ the pairs with product 5 are $\displaystyle 1$ , $\displaystyle 5$ and $\displaystyle -1$ , $\displaystyle -5$ : their sums are $\displaystyle 6$ and $\displaystyle -6$ , never $\displaystyle 4$ . Such a trinomial cannot be factored this way.

Exercise 3

Which of the trinomials can be factored with whole numbers?
Factor those, and explain your answer.

/ – 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

In the trinomial $\displaystyle x^{2}+\square x+12$ a whole number is missing.
Find every whole number that can stand in the box so that the trinomial can be factored with whole numbers.
How many are there?

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