Get ready for Precalculus
- 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)
When we are given the graph of a polynomial, we can deduce what its zeros are, which helps us determine a few factors the polynomial's equation must include.
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- how does the point: 1.5 make 3/2?(4 votes)
- How do you know whether the graph is upwards opening or downward opening, could you multiply the binomials, and then simplify it to find it?(4 votes)
- Typically when given only zeroes and you want to find the equation through those zeroes, you don't need to worry about the specifics of the graph itself — as long as you match it's zeroes.
However, if you want to graph the equation at it's entirety, the best way (imo, if someone else has a better option please leave down in the replies/comments for him), is to simply multiply it out.
* It is best to know that if it's a quadratic equation (degree is 2), just find whether or not a is positive or negative, open up or down at it's respective order.
* If it's a cubic or higher , it's better to refer to demos or graph by hand as there are multiple phases that the graph could potentially open up or down.
hopefully that helps !(8 votes)
- Is the concept of zeros of polynomials: matching equation to graph the same idea as the concept of the rational zero theorem?(3 votes)
- The concept of zeroes of polynomials is to solve the equation, whether by graphing, using the polynomial theorem, graphing, etc.(1 vote)
- I was wondering how this will be useful in real life. Does anyone have a good solution?(2 votes)
- Obviously, once you get to math at this stage, only a few jobs use them. I guess that since polynomials can make curves when put on a graph, it can be used for construction planning.(2 votes)
- what does p(x) mean(1 vote)
- That refers to the output of functions p, just like f(x) is the output of function f. Function p takes in an input of x, and then does something to it to create p(x). Functions can be called all sorts of names.(4 votes)
- I don't understand where the Y comes from? aren't we dealing with polynomials with the variable x ? why is y = p(x)(1 vote)
- P(x) or more commonly symbolized as F(x) simply represents the y-value at a given point. When finding the "zeros" or "x-intercepts" of a function/graph you are trying to find the values that would result in each factor equaling zero resulting in the entire equation being equal to zero. Lets use some of the factors from the video (x+4) and (x-3), when you set these factors separately equal to zero x+4=0 and x-3=0 and solve, you find that the x values must be equal to -4 and 3 respectively, which means at those x-values on the graph the equation/ p(x) will be equal to zero or the Y-VALUE will be equal to zero. This is true for each factor.
Hope this helps!(2 votes)
Quick Q, isn't B correct as well? Or is it that since D had more solutions - that is the correct answer?
**I did solve for B, although I did with only -4 and 3 but didn't account the 3/2 that Sir had found but is it still correct?(1 vote)
- how did u get 3/2(1 vote)
- Quite simple acutally. If you take a look, when the line intercepts the x axis, there is: -4, 1.5, and 3. Sal said 3/2 instead of 1.5 because 1.5 in fraction form is 3/2.(1 vote)
- [Instructor] We are asked, what could be the equation of p? And we have graph of our polynomial p right over here, you could view this as the graph of y is equal to p of x. So pause this video and see if you can figure that out. All right, now let's work on this together, and you can see that all the choices have p of x, in factored form where it's very easy to identify the zeros or the x values that would make our polynomial equal to zero. And we could also look at this graph and we can see what the zeros are. This is where we're going to intersect the x-axis, also known as the x-intercepts. So you can see when x is equal to negative four, we have a zero because our polynomial is zero there. So we know p of negative four is equal to zero. We also know that p of, looks like 1 1/2, or I could say 3/2. p of 3/2 is equal to zero, and we also know that p of three is equal to zero. So let's look for an expression where that is true. And because it's in factored form, each of the parts of the product will probably make our polynomial zero for one of these zeroes. So let's see if, if in order for our polynomial to be equal to zero when x is equal to negative four, we probably want to have a term that has an x plus four in it. Or we want to have a, I should say, a product that has an x plus four in it. Because x plus four is equal to zero when x is equal to negative four. Well we have an x plus four there, and we have an x plus four there. So I'm liking choices B and D so far. Now for this second root, we have p of 3/2 is equal to zero so I would look for something like x minus 3/2 in our product. I don't see an x minus 3/2 here, but as we've mentioned in other videos you can also multiply these times constants. So if I were to multiply, let's see to get rid of this fraction here, if I multiply by two this would be the same thing as, let me scroll down a little bit, same thing as two x minus three. And you could test that out, two x minus three is equal to zero when x is equal to 3/2. And let's see, we have a two x minus three right over there. So choice D is looking awfully good, but let's just verify it with this last one. For p of three to be equal to zero, we could have an expression like x minus three in the product because this is equal to zero when x is equal to three, and we indeed have that right over there. So choice D is looking very good. When x is equal to negative four, this part of our product is equal to zero which makes the whole thing equal to zero. When x is equal to 3/2, two x minus three is equal to zero which makes the entire product equal to zero. And when x minus, and when an x is equal to three, it makes x minus three equal to zero. Zero times something, times something is going to be equal to zero.