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# Zeros of polynomials introduction

The zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero. They are interesting to us for many reasons, one of which is that they tell us about the x-intercepts of the polynomial's graph. We will also see that they are directly related to the factors of the polynomial.

## Want to join the conversation?

• what is the point of finding the zeros of polynomials? Are there any specific usage for it?
• I think it's mainly to help graph the polynomial. Similar to how you would use the vertex and x-intercepts to help graph a parabola.
• is there a way to factorise a polynomial with multiple roots to obtain its zeros?
• yep,
i hope i understood your question,
did you know that the highest exponential power of the variable is an indication of the max. no. of a zeros that a polynomial has/ (well now you know)
example; x+5 (highest degree of the variable(x)=1)
max. no. of zeros is 1
x^2 + x + 5 (highest power of the variable
=2)
max. no. of zeros is 2
x^n + (x^n-1) + 9 (highest power of the
variable = n)
max. no. of zeros is n
So if we consider a polynomial in variable x of highest power 2 (guess how many zeros it has)
= 4x^2 + 14x + 6
steps; multiply the co-efficient of x ^2 and the constant~ 4*6 =24
factorise the obtained product(24) such a way that it's sum is equal to the co-efficient of x
~24 = 1*24,2*12,3*,4*6
but 2*12 =24 as well as 2+ 12=14 (the co-efficient of x)
rewrite the expression as
4x^2 +12x+2x + 6
= 4x(x+3) + 2(x+3)
=(x+3)(4x+2)
so the zeros of our polynomial are -3 and -2/4
similarly you can try converting a polynomial of different degrees to 2 by dividing.
and..............there is a relation between zeroes and the coefficients
for ax^2 + bx + c (a is not equal to zero)
the sum of zeroes = -(b)/a
product of zeroes = c/a
why? i recommend you to watch this video of link given ~ https://www.youtube.com/watch?v=aEDrlGFrsuo
``the video also gives info about the relations between zeroes and co-efficients of different types of polynomials``

this is how much i know till now,
hope this helps
chao
• How many polynomials can you have with -2 and -5 as its zeroes?
• Technically infinity, but also one at the same time...?

This is the main one:

x^2 + 7x + 10

but you can also have

2x^2 + 14x + 20
3x^2 + 21x + 30

etc. So I guess the way you would write this is

n(x^2 + 7x + 10)

where n is just some random number.

You also didn't specify that this has to be a trinomal, so there are a lot of others that also fit, like these:

x^3 + 9x^2 + 24x + 20
x^3 + 12x^2 + 45x + 50
x^4 + 14x^3 + 69x^2 + 140x + 100

So, to answer your question fully, there are an infinite amount of ways to make a polynomial have only -2 and -5 as its zeroes. Sorry for being kinda disorganized.
• Um, is there a point for this video? or is sal just showing us something interesting you can do with zeros?

thanks!

P.S. What's Desmos?
• This video is to introduce the topics of zeroes (which are the solutions) of a polynomial. Zeroes are essential to the foundation of algebra functions, so keep an eye about that !
Desmos is a free website where you can graph functions, here is the link:
https://www.desmos.com/calculator
This is very handy tool in virtual learning when you don't have a graphing calculator near you.
• If we write a polynomial in terms of y that is x = p(y) then will the point of intersection with y axis be considered as zeroes of the polynomial?
• the x and y are arbitrary, but if you want to keep the traditional x is the horizontal axis and y is the vertical axis, then they may not be actually called zeroes, but they have the same function.

You can think of it like having a normal y=f(x) but swapping the xs and ys, which in effect reflects the graph about the line y=x. In other words all x and y values swap.

It's worth noting that this x=p(y) can very likely not be a function any more. The simplest way to tell is if it does not cross the vertical line test.
• so in the graph that is shown dips right? its a curve. and while it does intercept the x axis at all the right places how do we know where exactly it intercepts the y axis? since it isn't mentioned , can't the curve "dip" anywhere and therefore intercept the y axis pretty much anywhere?
• yes, the curve is know as parabola. Actually yes, it can "dip" anywhere and intercept the y axis but it not important unless the equation is in y form. Here we have taken the equation in x form that is p(x)= 0.
so we form a equation with x and consider the value of y such that it turns the equation to zero. Also the y and x intercepts will be much clearer if you refer to The liner equations in two variables section of maths! ^^
• Are polynomials that hard?
• It depends on the person. If it seems to you as boring then its the toughest. If you think its easy then its easiest.
• Why is the line drawn curved why is it not a straight line?
• The reason the line is drawn curved rather than a straight line is because Sal only figured out the zeros of the polynomial. The zeros of the polynomial are only the x values that make the polynomial equals 0. If you took the time to graph out all the x points on the graph, it would show the line is curved rather then just a straight line. Hope this helps.
• should zeros be written as for example, x=2, or should it be written as (2,0)??
• These would be the cartesian coordinates of the roots, or the place where the polynomial intercepts the x axis. However, not all roots will be given, as some might be complex.