RELATIONSHIP BETWEEN THE ZEROS (ROOTS)
AND FACTORS OF POLYNOMIALS

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if  $\,x-\color{red}{2}\,$  is a factor,   then  $\color{red}{2}$  is a zero

if  $\color{red}{2}$  is a zero,   then  $\,x-\color{red}{2}\,$  is a factor

Linear Factors

Factors of the form $\,ax + b\,$ are called linear factors.
Here, $\,a\,$ and $\,b\,$ are real numbers, with $\,a\ne 0\,.$

In a linear factor, you must have a variable (say, $\,x\,$), which can only be raised to the first power.
No $x^2$, no $x^3$, no $x$ in denominators, no $x$ under square roots, and so on.
The variable may be multiplied by a nonzero real number, and there may be a constant term.

Thus, all the following are examples of linear factors:

• $2x + 3$     ($a = 2$,   $b = 3$)
• $2x - 3$     ($a = 2$,   $b = -3$)
• $\hphantom{2}x + 3$     ($a = 1$,   $b = 3$)
• $\hphantom{2}x - 3$     ($a = 1$,   $b = -3$)

In particular, note that:

Examples of Linear Factors of the form $\,x - c$

The table below also shows some alternate names for linear factors of the form $\,x - c\,$:

$c$	linear factor $x-c$	alternate name for $\,x-c$	note
$2$	$x-2$
$\displaystyle\frac 12$	$\displaystyle x-\frac 12$	$\displaystyle\frac 12(2x-1)$	factor out $\displaystyle\frac 12$
$-2$	$x-(-2)$	$x + 2$	if $\,c\,$ is negative, then the factor takes the form ‘$x + (\text{positive #})$’
$\displaystyle \frac 53$	$\displaystyle x-\frac 53$	$\displaystyle \frac 13(3x - 5)$	factor out $\displaystyle\frac 13$

Equivalent Characterizations of a Zero of a Polynomial

The previous section, Introduction to Polynomials, gave a list of equivalent ways to characterize a zero of an arbitrary function.
For polynomials, we can extend the list:

EXAMPLE

For each set of conditions given below, there is EXACTLY ONE, MORE THAN ONE, or NO polynomial $\,P\,$ that satisfies all the stated conditions.

• If no such polynomial exists, give a reason why.
• If exactly one polynomial exists, give a formula for the polynomial.
• If more than one polynomial is possible, give two different polynomials that satisfy all the stated conditions.

SET OF CONDITIONS	ANALYSIS	ANSWER
the degree of $\,P\,$ is $\,5\,$ $-3\,$ is a zero of $\,P$ $x - 2\,$ is a factor of $\,P(x)$ the graph of $\,P\,$ crosses the $x$-axis at $\,x = 4$	The conditions are not necessarily used in the order given. since $-3\,$ is a zero, $\,x + 3\,$ is a factor $\,x - 2\,$ is a factor since the graph of $\,P\,$ crosses the $x$-axis at $\,x = 4\,,$ the number $\,4\,$ is a zero, so $\,x-4\,$ is a factor at this point, we have $\,P(x) = (x+3)(x-2)(x-4)\,,$ which has degree $\,3\,$ to get degree $\,5\,,$ you can raise the existing factors to higher powers, and/or put in additional factors you can also include a constant factor	MORE THAN ONE polynomial is possible For example: $$\cssId{s101}{P(x) = (x+3)(x-2)^2(x-4)^2}$$ or $$\cssId{s103}{P(x) = 5(x+3)(x-2)(x-4)(x+1)(x-7)}$$ There are many other correct answers.
both $\,2\,$ and $\,3\,$ are roots of $\,P\,$ $x = 5\,$ is a solution of the equation $\,P(x) = 0\,$ the degree of $\,P\,$ is $\,2\,$	since $2\,$ is a zero, $\,x - 2\,$ is a factor since $3\,$ is a zero, $\,x - 3\,$ is a factor since $\,P(5) = 0\,,$ $\,x - 5\,$ is a factor at this point, we have $\,P(x) = (x-2)(x-3)(x-5)\,,$ which has degree $\,3\,$	NO polynomial is possible The stated conditions force the polynomial to have at least degree $\,3\,.$
$x - 1\,$ goes into $\,P(x)\,$ evenly $P\,$ has degree $\,4$ the points $\,(0,0)\,$ and $\,(-3,0)\,$ are on the graph of $\,P\,$ $P(2) = 3$ $7\,$ is a zero of $\,P$	$\,x - 1\,$ is a factor since $\,0\,$ is a zero, $\,x\,$ is a factor since $\,-3\,$ is a zero, $\,x + 3\,$ is a factor since $\,7\,$ is a zero, $\,x -7\,$ is a factor at this point, we have $\,P(x) = (x-1)(x)(x+3)(x-7)\,,$ which has degree $\,4$ the only possible remaining factor is a constant factor $\,K$ $\,P(x) = K(x-1)(x)(x+3)(x-7)\,$ $\,P(2) = K(2-1)(2)(2+3)(2-7) = -50K\,$ Since $\,P(2) = 3\,,$     $\,-50K = 3\,,$     $\displaystyle\,K = -\frac{3}{50}$	EXACTLY ONE polynomial satisfies the stated conditions $\displaystyle P(x) = -\frac{3}{50}x(x-1)(x+3)(x-7)$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
end behavior of polynomials