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### Course: Get ready for AP® Calculus > Unit 5

Lesson 6: Zeros of polynomials- Zeros of polynomials introduction
- Zeros of polynomials: plotting zeros
- Zeros of polynomials: matching equation to zeros
- Zeros of polynomials: matching equation to graph
- Zeros of polynomials (factored form)
- Zeros of polynomials (with factoring): grouping
- Zeros of polynomials (with factoring): common factor
- Zeros of polynomials (with factoring)

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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?(16 votes)
- 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.(26 votes)

- is there a way to factorise a polynomial with multiple roots to obtain its zeros?(10 votes)
- 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(8 votes)

- How many polynomials can you have
**with -2 and -5**as its zeroes?(5 votes)- 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.(11 votes)

- 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?(4 votes)- 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.(8 votes)

- 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?(4 votes)
- 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.(2 votes)

- so in4:39the 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?(3 votes)
- 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! ^^(4 votes)

- Are polynomials that hard?(2 votes)
- It depends on the person. If it seems to you as boring then its the toughest. If you think its easy then its easiest.(4 votes)

- Why is the line drawn curved why is it not a straight line?(3 votes)
- 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.(2 votes)

- should zeros be written as for example, x=2, or should it be written as (2,0)??(2 votes)
- 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.(2 votes)

- how can 3 be equal to zero if -2 + 3 equals 1?(1 vote)
- p(x) = (x-1)(x+2)(x-3)(x+4)

If you input 3 into all of the x's, you will get:

p(3) = (3-1)(3+2)(3-3)(3+4) = (2)(5)(0)(7) = 0.

Hope that helps!(2 votes)

## Video transcript

- [Instructor] Let's say that
we have a polynomial, p of x, and we can factor it, and we can put it in the form
x minus one times x plus two times x minus three times x plus four. And what we are concerned with are the zeros of this polynomial, and you might say, "What
is a zero of a polynomial?" Well, those are the x-values that are going to make the
polynomial equal to zero. So another way to think about it is, for what x-values is p of x
going to be equal to zero, or another way you can think about it is, for what x-values is this expression going to be equal to zero. So for what x-values is x
minus one times x plus two times x minus three times x plus four, going to be equal to zero. I encourage you to pause this video, and think about that a little bit before we work through it together. Well the key realization here
is if you have the product of a bunch of expressions, if any one of them is equal to zero, it doesn't matter what the others are, because zero times anything else is going to be equal to zero. So the fancy term for that
is the zero product property. But all it says is, hey,
if you can find an x-value that makes any one of these
expressions equal to zero, well that's going to make the
entire expression going to be, it's going to make the entire
expression equal to zero. So, the zeros of this polynomial
are gonna be the x-values that could make x minus one equal to zero. So x minus one equals zero. Well we know what x-value
would make that happen, if x is equal to one, if you
add one to both sides here, x equals one, so x equals one
is a zero of this polynomial. Another way to say that is
p of one when x equals one, that whole polynomial is
going to be equal to zero. How do I know that? Well if I put a one in, right over here, this expression right
over here, x minus one, that is going to be equal to zero. So you're gonna have zero
times a bunch of other stuff which is going to be equal to zero. And so by the same idea, we can figure out what
the other zeros are. What would make this part equal to zero? What x-value would make
x plus two equal to zero? Well, x equals negative
two, x equals negative two, would make x plus two equal zero. So x equals negative two is
another zero of this polynomial. And we could keep going. What would make x minus
three equal to zero? Well if x is equal to three, that would make x minus
three equal to zero, and that would then make
the entire expression equal to zero. And then last but not least, what would make x plus four equal to zero? Well if x is equal to negative four. And just like that we have found four zeros
for this polynomial, when x equals one the
polynomial's equal to zero, when x equals negative two the
polynomial's equal to zero, when x equals three the
polynomial's equal to zero, and when x equals negative four the polynomial's equal to zero. And one of the interesting things about the zeros of a polynomial you could actually use
that to start to sketch out what the graph might look like. So, for example, we know,
that this polynomial is going to take on the
value zero at these zeros. So let me just draw a rough
sketch right over here. So this is my x-axis, that's
my y-axis, that's my y-axis. And so, let's see, at x equals one, so let me just do it this way, so we have one, two, three, and four. Then you have negative one,
negative two, negative three, and then last but not
least, negative four. We know that this polynomial, p of x, is going to be equal to
zero at x equals one. So it's going to intersect
the x-axis right there. It's going to be equal to
zero at x equals negative two, so right over there. At x equals three, right over there. And x equals negative four. Now we don't know exactly
what the graph looks like, just based on this. We could try out some values
on either side to figure out, hey, is it above the
x-axis, or below the x-axis, for x-values less than negative four. And we can try things out like that but we know it intersects
the x-axis at these points. So it might look something like this, this is a very rough sketch. It might look something like this, we don't know without doing
a little bit more work. But ahead of time, I took a
look at what this looks like, I went on to Desmos, and I graphed it, and you can see, it looks
exactly as what we would expect. The graph of this polynomial
intersects the x-axis at x equals negative four,
actually let me color code it, x equals negative four, and that is that zero right over there, x equals negative two,
that's this zero right there, x equals one, right over there, and then x equals three, right over there. In future videos we will
study this in even more depth.