Asked • 07/01/20

It's all about Polynomials!

Suppose you have a third-degree Polynomial with all (4) coefficients greater than 0:


  1. How many x-intercepts do you have?
  2. How many y-intercepts do you have?
  3. What are the signs of the intercepts?
  4. Under what condition is there no (local) min or max?
  5. What is the sign of the x-coordinate of the point of inflection?

Joel L.

tutor
There is a flaw in the question. A third degree polynomial can't have 4 coefficients. To define it mathematically, A coefficient is a number or quantity placed (generally) before and multiplying another quantity, as 3 in the expression 3x. So if I have a polynomial 6x^3 + 4x^2 + 5x +7, the numbers 6, 4 and 5 are coefficients and 7 is a constant term. So if you meant 3 coefficients and a constant term, each greater than 0, then you can refer to the answers our expert tutors.
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07/01/20

Egbert M.

The constant is the coefficient of the exponential x^0. A polynomial of n-th degree has the form [sum (ai * x^i)], where i runs through the range from 0 to n.
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07/01/20

Joel L.

tutor
Sorry but I disagree with you Egbert. That is still called constant not coefficient according to the definition. x itself is a variable but if you put raise to 0 (zero) that becomes constant that is equal to 1.
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07/02/20

Joel L.

tutor
Another thing, you mentioned 3 is the coefficient of x^0. How about if x=0? So, you say 3*0^0=3?
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07/02/20

2 Answers By Expert Tutors

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Mark M. answered • 07/01/20

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Mathematics Teacher - NCLB Highly Qualified

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By the Fundamental Theorem of the Algebra, a polynomial of degree n as n roots.

These roots may or may not be real. The real roots shall be zeros or x-intercepts.


A function, by definition, has only one y-intercept.


The signs of the x-intercept can be determined using Descartes Rule of Signs.


The sign of the y-intercept is determined by the constant term of the polynomial.


The point of inflection is at -b / 2a of the first derivative of the funcion.

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Egbert M. answered • 07/01/20

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Precalculus is the most important "next step" in Math after Algebra!

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  1. A function can only have one x-intercept!
  2. This polynomial only has one y-intercept!
  3. The x-intercept is negative, the y-intercept is positive
  4. There is no min or max if b2 = 3ac, they "merge" at the point of inflection!
  5. The point of inflection is at -b/3a
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