A polynomial as a product of factors

Calculus, grade 10 (5 units) Lesson 3 of 26Learning

A polynomial as a product of factors

The zeros from the factors

When a polynomial is written as a product of factors, its zeros show at once: a product is 0 only when one of its factors is 0.

The example below reads $\displaystyle {f(x)}={(x+2)(x-1)(x-3)}$ this way:

The zeros: $\displaystyle {x}={-2}$ , $\displaystyle {x}={1}$ , $\displaystyle {x}={3}$ .
Multiplying out, the term with the highest power will be $\displaystyle {x\cdot x\cdot x}={x^3}$ : degree three, with a positive coefficient. So the left end of the graph points down, and the right end up.
At each zero one factor changes sign, so the product changes sign too:

$\displaystyle x$	$\displaystyle x<-2$	$\displaystyle -2<x<1$	$\displaystyle 1<x<3$	$\displaystyle x>3$
$\displaystyle f(x)$	−	+	−	+

The multiplicity of a zero

A factor may come with a power. In $\displaystyle (x-1)^2$ the zero $\displaystyle {x}={1}$ has multiplicity 2, and in $\displaystyle (x-1)^3$ – multiplicity 3.

Odd multiplicity (1, 3 …): the factor changes sign, and the graph crosses the $\displaystyle x$ -axis.
Even multiplicity (2, 4 …): the factor is not negative on either side, and the graph only touches the $\displaystyle x$ -axis and turns back.

On the board below, $\displaystyle {f(x)}={(x+2)(x-1)^2}$ : at $\displaystyle {x}={-2}$ the graph crosses the axis, and at $\displaystyle {x}={1}$ it touches it. The function is negative where $\displaystyle x<-2$ , and positive where $\displaystyle -2<x<1$ and also where $\displaystyle x>1$ .

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

A quadratic factor with no roots

A factor like $\displaystyle x^2+1$ or $\displaystyle x^2+x+1$ is never 0 (its discriminant is negative), so it is positive for every $\displaystyle x$ .

Such a factor adds no zeros and changes no sign.
It does change the degree and the shape of the graph.
$\displaystyle {f(x)}={(x-2)(x^2+1)}$ has one zero, $\displaystyle {x}={2}$ : it is negative where $\displaystyle x<2$ and positive where $\displaystyle x>2$ , like $\displaystyle x-2$ .

A sketch of a polynomial

The factors alone are enough to sketch the graph, in four steps:

The zeros and the multiplicity of each.
The term with the highest power: where the ends point.
A table of signs.
A line through the zeros, crossing or touching by the multiplicity, that matches the table.

The example below sketches $\displaystyle {f(x)}={x(x+2)(x-2)^2}$ :

The zeros: $\displaystyle {x}={-2}$ and $\displaystyle {x}={0}$ with multiplicity 1 (crossing), $\displaystyle {x}={2}$ with multiplicity 2 (touching).
The highest power: $\displaystyle {x\cdot x\cdot x^2}={x^4}$ – an even degree and a positive coefficient: both ends point up.
The signs: positive where $\displaystyle x<-2$ , negative where $\displaystyle -2<x<0$ , and positive where $\displaystyle 0<x<2$ and where $\displaystyle x>2$ .

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

The graph on the board matches all the steps. The factors do not tell exactly where the dip between $\displaystyle -2$ and 0 and the peak between 0 and 2 are.

What the factors do not tell

The factors tell how many extremum points there are at least, but not where they are:

Between two neighbouring zeros the graph leaves the axis and comes back to it, so there is at least one extremum point between them.
At a zero of even multiplicity there is an extremum point: the graph touches the axis and turns back.
Not every polynomial factors easily: $\displaystyle x^3+x+1$ has a zero between $\displaystyle -1$ and 0, but it is not a number that is easy to find.

Finding the extremum points exactly needs a new tool – the derivative, which comes later in the course.

Exercise 1

The function is $\displaystyle {f(x)}={(x-1)(x+3)(x-4)}$ .

Find the zeros of $\displaystyle f$ .

Where do the ends of the graph point?
Make a table of signs of $\displaystyle f$ .

Mark the zeros on the graph, and check your table.

\displaystyle f(x)=(x-1)(x+3)(x-4)

Exercise 2

The board below shows the graph of a polynomial of degree three.
The coefficient of $\displaystyle x^3$ is 1.
The graph meets the $\displaystyle y$ -axis at the point $\displaystyle (0,4)$ .
Write the function in a row of the board, as a product of factors.

\displaystyle f(x)=\,?

\displaystyle A=(0,4)

Exercise 3

The function is $\displaystyle {f(x)}={x^3-4x}$ .

Factor $\displaystyle f(x)$ .

/ – 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

Find the zeros of $\displaystyle f$ .

Make a table of signs of $\displaystyle f$ .
Is $\displaystyle f$ even, odd or neither?
How does the table show it?

Exercise 4

The function is $\displaystyle {f(x)}={(x^2+4)(x-3)}$ .

Find the zeros of $\displaystyle f$ .

Explain why $\displaystyle f$ has only one zero, although its degree is three.
Write where $\displaystyle f$ is positive and where it is negative.

Exercise 5

The board below shows the graph of a polynomial of degree three.
The coefficient of $\displaystyle x^3$ is $\displaystyle -1$ .
Write the function in a row of the board, as a product of factors.

\displaystyle f(x)=\,?

Exercise 6

The function is $\displaystyle {f(x)}={-(x+1)^2(x-2)(x-4)}$ .
Answer without multiplying out:

What is the degree of $\displaystyle f$ , and where do the ends of the graph point?
At which zeros does the graph cross the $\displaystyle x$ -axis, and at which does it touch it?
Make a table of signs of $\displaystyle f$ .
How many extremum points does $\displaystyle f$ have at least? Explain.

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
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 properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

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

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
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 instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
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
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

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
$\displaystyle y-y_1=m(x-x_1)$

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
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$