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# Polynomial factors and graphs — Harder example

Watch Sal work through a harder Polynomial factors and graphs problem.

## Want to join the conversation?

• find a harder example. This is way too easy when all you have to do is count the x intercepts and how many times the graph touches the xaxis • what is the difference between distinct zero and double zero?Why would the graph bounce back?What is the official definition of double zero?Thanks. • A double zero results from a function having a repeated root, for example: roots derived from factors of the form (x-a)^2. We already know that roots occur where the graph touches/cuts the x axis, so if a factor is of `some squared form` then the corresponding y values of the function would be `positive.` At the point of the root, the graph `doesn't cross the x axis` (because a quadratic function governs that portion of the graph) but instead bounces back from the x axis. Get it?
• What's the difference between distinct and double zeros? • Can’t i just apply the rule that if x power’s even then both negative and positive curves goes in the same direction and if it is negatives then doesn’t • Just look at the exponents if the exponents are even the graph lines go to the same direction
Since the graph has both arrows down the exponent SHOULD be EVEN • Can you find a harder question like from a practice test or something? The hard questions aren't supposed to be this easy. • I don't get why A does not work. Something about end behavior. • You can explain why A doesn't work 2 ways. You could talk about end behavior: You should know that the end behaviour of polynomials is determined by if their degree is an odd or even power: if odd, then the ends go in opposite directions and if even, the ends goin the same direction. Here, our ends are going in the same direction so we need to pick something with an even degree. Adding up all the x's you see, for A) we get a degree of 3 and for B) we get 4. This allows us to rule out A).
Another way you can approach this is by thinking about multiplicity of roots, or how often the root is repeated (or when the polynomial is shown in root form like here, what the exponent on each root is). Roots with multiplicity 1 have the graph passing through them, roots with even multiplicity have the graph touch the root and "bounce" backwards, and roots with odd multiplicity greater than 1 have the graph do a little "squiggle", where it becomes horizontal at the root but then eventually crosses it. Here, at x=0 the graph goes down to 0 but then bounces back, so we need an even multiplicity for that root, which A) does not have and B) does have.
• one can also use the odd and even aspect right? Like X raised to 1 which is odd hence shud face diff directions and X raised to 2 which is even facing same direction downwards. • can't understand the difference between choice a) & b) • A simple way of looking at this would be to remember that any number squared is positive.
So, in option B, the first x value (before negation; x^2), whether initially positive or negative, will always be positive. That number is then negated (-x^2), and so will always be a negative number. (And would show a downward trend, which is what the graph displays).

However, in choice A, a negative x value would immediately become a positive number, as any number negated twice is positive. (And would show an upward trend, which is not what the graph displays.)

Hope that helps, and feel free to ask if anything is unclear! 