Find Polynomial of Degree n...1

Section 2.2
Question 66

Can you solve 66 as a guide for me to do a few more?

 

66.
x=-2,6

to find a polynomial use zero product rule
since given two roots, x[1]=-2 and x[2]=6, you need

f(x)=(x-x[1])(x-x[2])

f(x)=(x-(-2))(x-6)
f(x)=(x+2)(x-6)
f(x)=x^2 - 4x - 12-> your polynomial

 

Easy stuff.
What is the zero product rule?

 

The "Zero Product Property" says that:

If a *b = 0 then a = 0 or b = 0 (or both a=0 and b=0).


It can help us solve equations:

Example: Solve (x−5)(x−3) = 0

The "Zero Product Property" says:

example:

If (x−5)(x−3) = 0 then (x−5) = 0 or (x−3) = 0

Now we just solve each of those:

For (x−5) = 0 we get x = 5

For (x−3) = 0 we get x = 3

And the solutions are: x = 5, or x = 3

Here it is on a graph:



y=0 when x=3 or x=5

another example:

3(x - 2) = 3x(x - 2)

Use "Standard Form":
3(x - 2) - 3x(x - 2) = 0...........factor completely

Which can be simplified to:
(3 - 3x)(x - 2) = 0
or 3(1 - x)(x - 2) = 0 ...........where 3, (1 - x) and (x - 2) are factors

Then the "Zero Product Property" says:

since 3>0, then zeros are x values that makes

(1 - x) = 0, or (x - 2) = 0

so, x = 1 or x = 2

 

A clear and precise reply.