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### Course: Digital SAT Math>Unit 8

Lesson 13: Polynomial and other nonlinear graphs: medium

# 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
• the exercises are then going to be much more difficult
• 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
• 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?
• 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
• What's the difference between distinct and double zeros?
• When the graph bounces like on the last choice, both those zeros are actually double zeros. When you solve for x values when y=0, you will have two identical answers for every time the graph bounces back.
• 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.
• 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.