Question 282871: the question is:find all the zeros of the function.is there a relationship between the number of real zeros and the number of x-intercepts of the graph? explain.f(x)=x^3-4x^2+x-4
this is what I have so far and I don't know were to go from here.
(x^3-4x^2)+(x-4)=x^2(x-4)+(x-4)
Answer by nyc_function(2741)   (Show Source):
You can put this solution on YOUR website! To find the zeros means to find the number(s) that when replaced for the given x in your polynomial function, will create zero for the function. In other words, we need to find numbers (or maybe just one number) that when used in place of every x in your function, zero is the result.
We can use synthetic division to find our zero(s).
See the video clip to learn about synthetic division.
http://www.youtube.com/watch?v=HY2UylGTDYU
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Back to your question.
I will use synthetic division on paper until finding one number or group of numbers that will give me a zero remainder.
List the factors of the leading coefficient and the constant.
Factor of 1 = 1
Factors of 4 = 1, 2, and 4
We now divide the constant's factors by the factor of the leading coefficient.
±1/1, ±2/1 and ±4/1 = ±1, ±2 and ±4
After using synthetic division, I found the only zero to be x = 4.
In other words, if you replace x with 4 in your function, you will get zero on both sides of the equation.
Why don't you try it?
0 = (4)^3 - 4(4)^2 + 4 - 4...After doing the math, the right side should also produce zero.
I'll let you do the math.
The x-intercept of any function is the location where the graph of the function crosses the x-axis. So, there is a relationship between the x-intercepts and the zeros. In fact, the x-intercepts = the roots = the solution = the zeros.
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