Algebraic technique: factoring, powers and roots

Algebra and calculus, grade 10 (4 units) Lesson 1 of 20Learning

Algebraic technique: factoring, powers and roots

Factoring

Factoring turns a sum into a product. A product is 0 only when one of its factors is 0 – and that is how, later in the course, you will find where a derivative is 0 and where a function changes sign.

Three ways to factor:

A common factor: $\displaystyle {3x^2-12x}={3x(x-4)}$ .
The short multiplication formulas: $\displaystyle {x^2-10x+25}={(x-5)^2}$ , and $\displaystyle {4x^2-9}={(2x+3)(2x-3)}$ .
Factoring a trinomial – in the next box.

The short multiplication formulas, in the factoring direction:

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

\displaystyle =

\displaystyle (a-b)^2

\displaystyle a^2-b^2

\displaystyle =

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

Factor to the end: after taking out a common factor, check whether what is left in the brackets factors further.

2x^3-8x=2x(x^2-4)=2x(x-2)(x+2)

Factoring a trinomial

A trinomial $\displaystyle ax^2+bx+c$ factors by its roots: $\displaystyle x_1$ and $\displaystyle x_2$ are the solutions of $\displaystyle {ax^2+bx+c}={0}$ .

\displaystyle ax^2+bx+c

\displaystyle =

\displaystyle a(x-x_1)(x-x_2)

When $\displaystyle {a}={1}$ , look for two numbers whose product is $\displaystyle c$ and whose sum is $\displaystyle b$ . In $\displaystyle x^2-7x+12$ : the product is 12 and the sum $\displaystyle -7$ – these are $\displaystyle -3$ and $\displaystyle -4$ , so $\displaystyle {x^2-7x+12}={(x-3)(x-4)}$ .
Any other trinomial: find its roots by the quadratic formula.

The example below factors $\displaystyle 2x^2+5x-3$ . By the quadratic formula:

x_{1,2}=\frac{-5\pm\sqrt{25+24}}{4}=\frac{-5\pm7}{4}

$\displaystyle {x_1}={\frac{1}{2}}$ , $\displaystyle {x_2}={-3}$ .
$\displaystyle {2x^2+5x-3}={2\left(x-\frac{1}{2}\right)(x+3)}={(2x-1)(x+3)}$ – the 2 went into the first brackets.

Algebraic fractions

An algebraic fraction is reduced like a fraction of numbers: factor the numerator and the denominator, and cancel a factor that appears in both.

\frac{x^2-4x+4}{x^2-4}=\frac{(x-2)^2}{(x-2)(x+2)}=\frac{x-2}{x+2}

The fraction is defined only where the denominator is not 0: here $\displaystyle x\neq2$ and $\displaystyle x\neq-2$ . Write this condition before reducing, because after it the factor $\displaystyle x-2$ no longer shows in the denominator.
To add or subtract, bring the fractions to a common denominator – it too comes from factoring:

\frac{x}{x-2}-\frac{4}{x^2-2x}=\frac{x\cdot x}{x(x-2)}-\frac{4}{x(x-2)}=\frac{x^2-4}{x(x-2)}=\frac{(x-2)(x+2)}{x(x-2)}=\frac{x+2}{x}

The laws of powers

The laws of powers from grade 9, for the same base:

\displaystyle a^m\cdot a^n

\displaystyle =

\displaystyle a^{m+n}

\displaystyle \frac{a^m}{a^n}

\displaystyle =

\displaystyle a^{m-n}

\displaystyle (a^m)^n

\displaystyle =

\displaystyle a^{mn}

\displaystyle (ab)^n

\displaystyle =

\displaystyle a^nb^n

The example below simplifies an expression with three of the laws:

\frac{(2x^3)^2}{4x^4}=\frac{4x^6}{4x^4}=x^2

Square roots

A product of roots is the root of the product, and a quotient of roots is the root of the quotient:

\displaystyle \sqrt{a}\cdot\sqrt{b}

\displaystyle =

\displaystyle \sqrt{ab}

\displaystyle a\geq0,\ b\geq0

\displaystyle \frac{\sqrt{a}}{\sqrt{b}}

\displaystyle =

\displaystyle \sqrt{\frac{a}{b}}

\displaystyle a\geq0,\ b>0

Taking a factor out of a root: write the number under the root as a product with a square in it. $\displaystyle {\sqrt{50}}={\sqrt{25}\cdot\sqrt{2}}={5\sqrt{2}}$ .
Putting a factor into a root: the factor goes in as its square. $\displaystyle {3\sqrt{2}}={\sqrt{9}\cdot\sqrt{2}}={\sqrt{18}}$ .
Roots add only when the numbers under them are the same: $\displaystyle {\sqrt{8}+\sqrt{2}}={2\sqrt{2}+\sqrt{2}}={3\sqrt{2}}$ .

Exercise 1

Factor, until no factor can be factored further.

/ – 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 2

Reduce the fractions.

/ – 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

For which values of $\displaystyle x$ is the fraction in part 1 not defined?

Exercise 3

Write as one fraction, reduced.

/ – 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

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

Which number is greater: $\displaystyle 3\sqrt{2}$ or $\displaystyle 2\sqrt{5}$ ?
Put the factor into the root in each of them, and compare.

Formulas: algebra and calculus, grade 10 (4 units)

Open as a page

Algebraic technique

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
$\displaystyle (a+b)(a-b)$ $\displaystyle =$ $\displaystyle a^2-b^2$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$
The roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
The product of roots
$\displaystyle \sqrt{a}\cdot\sqrt{b}$ $\displaystyle =$ $\displaystyle \sqrt{ab}$ $\displaystyle a\geq0,\ b\geq0$
The quotient of roots
$\displaystyle \frac{\sqrt{a}}{\sqrt{b}}$ $\displaystyle =$ $\displaystyle \sqrt{\frac{a}{b}}$ $\displaystyle a\geq0,\ b>0$

The properties of a function

An even function – its graph symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – its graph symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The $\displaystyle x$ -coordinate of the vertex of $\displaystyle {y}={ax^2+bx+c}$
$\displaystyle x=\frac{-b}{2a}$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transforming a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units to the right ( $\displaystyle p<0$ – to the left)
$\displaystyle y=f(x-p)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$
A stretch ( $\displaystyle |a|>1$ ) or a shrink ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$

The derivative and the rules of differentiation

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the chord
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The derivative of a product
$\displaystyle (f(x)\cdot g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a composite function
$\displaystyle (f(g(x)))'$ $\displaystyle =$ $\displaystyle f'(g(x))\cdot g'(x)$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

The derivative of a square root function

The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$