Polynomial factors and graphs — Basic example
About this video
This algebra lesson uses graphs to decide which polynomial could have four distinct zeros. It explains that a zero is an x-value where the graph meets the x-axis, so four distinct zeros means exactly four x-axis crossings. It rules out graphs with three or five crossings and checks the remaining graph by counting.
You'll be able to
- define a zero of a polynomial as an x-intercept.
- count distinct zeros from a graph's x-axis crossings.
- rule out graphs that do not match a stated number of zeros.
Key words
- Zero of a polynomial
- An x-value at which the polynomial equals zero, which is where its graph meets the x-axis.
- Distinct zeros
- Zeros with different x-values; repeated zeros at the same x-value count once.
- X-intercept
- A point where the graph crosses the x-axis, with y equal to zero.
- Polynomial graph
- The graph of y = p(x), whose shape reveals its zeros.
Transcript
- [Instructor] The polynomial p of x has four distinct zeros, four distinct zeros. Which of the following graphs could represent y is equal to p of x. Well, four distinct zeros means that the graph is going to intersect the x-axis exactly four times. So this first one intersects the x-axis one, two, three times.
So we can rule that one out. This one intersects the x-axis one, two, three, four, five times. Let's go and rule that one out. This one intersects the x-axis one, two, three, four times. It has four distinct zeros. So this looks like, this one right over here looks like our choice, and this one right over here, it actually only has two distinct zeros, or at least that's what it looks like.
So I'd rule that one out as well. So this is definitely going to be the one that has four distinct zeros. Zeros are the x values at which the function is equal or the polynomial, in this case, which is a function of x, is going to be equal to zero, and you can see that's happening exactly four times.
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