Priya wants to sketch a graph of the polynomial defined by .
She knows , so she suspects that could be a factor of and writes and draws a diagram.
-1
Finish Priya’s diagram.
Write as the product of and another factor.
Write as the product of three linear factors.
Draw a sketch of .
12.3
Activity
Here are some polynomial functions with one or more known factors. Rewrite each polynomial as a product of linear factors.
Note: you may not need to use all the columns in each diagram. For some problems, you may need to make another diagram.
,
0
-7
,
,
3
, ,
(Hint: )
, , ,
Student Lesson Summary
What are some things that could be true about the polynomial function defined by if we know ?
Thinking about the graph of the polynomial, the point must be on the graph as a horizontal intercept.
Thinking about the expression written in factored form, could be one of the factors, since when .
How can we figure out whether actually is a factor?
If we assume that is a factor, then there is some other polynomial where , , and are real numbers and . In the past we have expanded to find . Instead, we can work out the values of , , and by thinking through the calculation backward.
One way to organize our thinking is to use a diagram. First, fill in and the leading term of , . From this we can see the leading term of must be , meaning , since .
We can fill in the rest of the diagram using similar thinking and paying close attention to the signs of each term. For example, we put in a in the bottom left cell because that’s the product of and . But that means we need to have a in the middle cell of the middle row, since that’s the only other place we will get an term, and we need to get once all the terms are collected. Continuing in this way, we get the completed table:
+24
Collecting all the terms in the interior of the diagram, we see that , so . Notice that the 24 in the bottom right was exactly what we needed, and it is how we know that is a factor of . With a bit more factoring, we can say that .
