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### Course: Integrated math 3>Unit 4

Lesson 2: Positive and negative intervals of polynomials

# Zeros of polynomials (multiplicity)

Given the graph of a polynomial and looking at its x-intercepts, we can determine the factors the polynomial must have. Additionally, we can determine whether those factors are raised to an odd power or to an even power (this is called the multiplicity of the factors).

## Want to join the conversation?

• What is the real life implement of such types of problems.
Why do we need to study these problems.
My curious mind always thinks about it.
• A curious mind is key to unlock the meaning of mathematics. Of course, I cannot assume what you want to be, that's your decision.

Polynomial roots can be found in the real life through data analysis of economics, sciences such as chemistry and physics, calculating costs of mortage balance, and much more. You have lots to find, as I cannot list all the ways that polynomial roots--rather, polynomials in general--in one post. Hopefully that helps !
• Okay, so just to summarize, if a line appears to "Bounce" on the x-axis, then the multiplicity is equal to an even number? Furthermore, the higher the multiplicity, the flatter the transition for an odd multiplicity and the narrower the bounce for an even multiplicity. is that a good summary of the video's content?
• What do you mean by sign change tho
• "Sign Change" means the signs of the terms switch from positive to negative or negative to positive. For example, in f(x)=-3x^2 +2x -4, there are 2 sign changes: -3x^2 to +2x, then +2x to -4. Hope this helps.
• How do you find the multiplicity and zeros of a polynomial?
• Wherever the polynomial touches the X axes a Zero is located there. Multiplicity is the type of polynomial that touches the x axis
• Is very confusing because in the video they explain they explain when a number is odd doesnt have a exponent but in the exercise they show as it should have
• if there is sign change, the exponent of that expression of the root would be raised to an odd power it doesn't necessarily have to be raised to the power of 1, it could be 3 or 7 or any odd number.

root of -3, sign change around it : (x+3)^odd exponent

if there is no sign change, expression will be raised to an even power

root of 2, no sign change : (x+2)^even exponent
• What is the numbers of even and odd change?
• What does Sal mean when he says the sign changes?
• By sign change, he mans that the Y value changes from positive to negative or vice versa. For example, if you just had (x+4), it would change from positive to negative or negative to positive (since it is an odd numbered power) but (x+4)^2 would not "sign change" because the power is even
• What do you mean "when you have a sign change" in the graph? How can you find that?
• A “sign change” is when the output of a function changes from positive to negative or vice versa.

Graphically, a sign change is when a function crosses the x-axis. To find where a function crosses the x-axis, solve for 0. To make sure the function is crossing the x-axis (and not just bouncing off) test an input above the x-intercept and below the x-intercept and make sure one is positive and one is negative.

Hope this helps!
• What happens to the point on the graph where x=0? Do we exclude any sign change here?
• No, sign changes only occur when the y-value switches from positive to negative or vice versa. You may have a sign change if y=0, x can =0 as well but y must.
• Would y = c (parallel to x-axis) have any zero (real or complex)?
(if no,
(in our textbook) it is given that a polynomial must have at least 1 zero, (not necessarily real nor distinct), is this statement false?)