In the prior section, Introduction to Polynomials, you began to see the beautiful relationship between the zeros and factors of polynomials. This section continues the discussion.
Suppose
[beautiful math coming... please be patient]$\,x2\,$ is a factor of a polynomial $\,P(x)\,$.
Then,
[beautiful math coming... please be patient]$\,P(x) = (x2)(\text{stuff})\,$.
Thus, $\,P(2) \ =\ (22)(\text{stuff}) \ =\ 0\cdot\text{stuff} \ =\ 0\,$.
Thus, $\,2\,$ is an input, whose corresponding output is zero.
Thus, $\,2\,$ is a zero of $\,P\,$.
So:
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:
The table below also shows some alternate names for linear factors of the form $\,x  c\,$:
$c$  linear factor $xc$ 
alternate name for $\,xc$ 
note 
$2$  $x2$  
$\displaystyle\frac 12$  $\displaystyle x\frac 12$  $\displaystyle\frac 12(2x1)$  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 $\displaystyle\frac 13$ 
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:
For each set of conditions given below, there is EXACTLY ONE, MORE THAN ONE, or NO polynomial $\,P\,$
that satisfies all the stated conditions.
SET OF CONDITIONS  ANALYSIS  ANSWER 

The conditions are not necessarily used in the order given.

polynomial is possible For example: $$P(x) = (x+3)(x2)^2(x4)^2$$ or $$P(x) = 5(x+3)(x2)(x4)(x+1)(x7)$$ There are many other correct answers. 


is possible The stated conditions force the polynomial to have at least degree $\,3\,$. 


satisfies the stated conditions $\displaystyle P(x) = \frac{3}{50}x(x1)(x+3)(x7)$ 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
