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 $\,x-2\,$ is a factor of a polynomial $\,P(x)\,.$
Then, $\,P(x) = (x-2)(\text{stuff})\,.$
Thus,
$\,\cssId{s6}{P(2)}
\cssId{s7}{\ =\ (2-2)(\text{stuff})}
\cssId{s8}{\ =\ 0\cdot\text{stuff}}
\cssId{s9}{\ =\ 0}\,.$
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 $x-c$ |
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 $\displaystyle\frac 13$ |
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 |
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The conditions are not necessarily used in the order given.
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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. |
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is possible The stated conditions force the polynomial to have at least degree $\,3\,.$ |
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satisfies the stated conditions $\displaystyle P(x) = -\frac{3}{50}x(x-1)(x+3)(x-7)$ |
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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