Finding Zeros Graphically
Finding Zeros by factoring
Polynomial Long Division
Intermediate Value Theorem
Miscellaneous
100
If f is a polynomial. Which special points on its graphs will help you to find its zeros?
The X-intercepts
100
My polynomial has 4 as a zero. What is one of its factors?
(x-4)
100
Divide x^2-9x-10 by x+1
x-10
100
Why can't a bird and a fish get married?
They have no place to make a home!
100
Determine the left and right behavior of the graph of 10x^7-3x^5+10x
Falling to the left and rising to the right.
200
Sketch a graph of a polynomial that has two zeros. (Many possible answers)
Sample answer: a parabola whose vertex is below the x axis.
200
If (2x+3) is a factor of f(x) then what is one of the roots of f(x)?
3/2
200
What is the remainder when x^3-x+5 is divided by x-2
7
200
In order to use the intermediate value theorem we need the graph of a polynomial to have two points with which properties?
One point has a positive y value and one has to have negative y value.
200
What types of asymptotes does the graph of 1/x have?
A horizontal and a vertical
300
What are the possible number of zeros that a parabola could have?
0, 1, 2
300
Find the roots of (x-2)^3(x+1)
2 and -1
300
Use the remainder theorem to evaluate the function f(x)=x^3-x+5 at 2
7
300
If f(2)=6 and f(6)=-3 what can you conclude from the Intermediate value theorem?
There is a root between 2 and 6.
300
What are the possible rational roots of 3x^2+5x+10
(positive and negative versions of:) 1, 2, 5, 10, 1/3, 2/3, 5/3, 10/3
400
Draw a polynomial graph which has no real zeros.
Many answers possible, example: a horizontal line.
400
Find the roots of x^2-2x-8
4, 2
400
Divide 4x^4 + 3x^3 + 2x + 1 by x^2 + x + 2
4x^2-x-7+ (11x+15)/(x^2+x+2)
400
Use the intermediate value theorem to show that -x^3+2x has a root between -1 and -2.
f(-1)=-1<0 and f(-2)=4<0
500
Explain why finding a zero is the same as finding an x-intercept.
A zero is a value a, where f(a)=0. An x intercept is a point on the graph (a, 0) where 0=f(a) since the point is on the graph.
500
If (x-a) is a factor of f(x), explain why this implies that a is a root of f(x).
This is because if (x-a) is a factor of f(x), f(x) is a product of (x-a) with other terms. When we plug in a for x, this factor becomes zero so the whole product is zero. Hence f(a)=0 and this, by definition, means a is a root.
500
Why doesn't the IVT work if the graph of a function is not continuous? (Draw a picture to explain)
Because we could hop over the x axis all together and never cross it. For example, the graph of 1/x.
500
Find the horizontal and vertical asymptotes of (x^2+3x+1)/(4x^2-9)
Horizontal: 1/4 Vertical 3/2 and -3/2
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