Synthetic Division...1

[nycmathguy]

College Algebra
Section R.6

 

[MathLover1]

correct

 

[nycmathguy]

correct

I like synthetic division.

Question:

Why does synthetic division only work when dividing by (x - a) or (x + a), where "a" is any integer?

 

[MathLover1]

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case.
You can only use synthetic division when you are dividing by something in the form of x±n.
If you are given, say, the polynomial equation y = x^2 + 5x + 6, you can factor the polynomial as y = (x + 3)(x + 2). Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x = –2 and x = –3 are the two zeroes of y.

You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x = –2 and x = –3 are the zeroes of a quadratic, then you know that x + 2 = 0, so x + 2 is a factor, and x + 3 = 0, so x + 3 is a factor. Therefore, you know that the quadratic must be of the form y = a(x + 3)(x + 2)

 

[nycmathguy]

Synthetic division is a shorthand, or shortcut, method of polynomial division in the special case of dividing by a linear factor -- and it only works in this case.
You can only use synthetic division when you are dividing by something in the form of x±n.
If you are given, say, the polynomial equation y = x^2 + 5x + 6, you can factor the polynomial as y = (x + 3)(x + 2). Then you can find the zeroes of y by setting each factor equal to zero and solving. You will find that x = –2 and x = –3 are the two zeroes of y.

You can, however, also work backwards from the zeroes to find the originating polynomial. For instance, if you are given that x = –2 and x = –3 are the zeroes of a quadratic, then you know that x + 2 = 0, so x + 2 is a factor, and x + 3 = 0, so x + 3 is a factor. Therefore, you know that the quadratic must be of the form y = a(x + 3)(x + 2)

Thanks for the information. Asyou know, we are currently in the review section of the textbook. We haven't done college algebra. I need to practice finding a polynomial function given the zeroes. I think there is a chapter somewhere down the line in this respect.

 

[MathLover1]

here are some links for you to practice finding a polynomial function given the zeroes:

https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-zeros-of-polynomials/
https://mcckc.edu/tutoring/docs/br/math/functions/Finding_the_Equation_of_a_Polynomial_Function.pdf

https://www.varsitytutors.com/algebra_ii-help/write-a-polynomial-function-from-its-zeros

https://tutorial.math.lamar.edu/problems/alg/FindingZeroesOfPolynomials.aspx

 

[nycmathguy]

here are some links for you to practice finding a polynomial function given the zeroes:

https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-zeros-of-polynomials/
https://mcckc.edu/tutoring/docs/br/math/functions/Finding_the_Equation_of_a_Polynomial_Function.pdf

https://www.varsitytutors.com/algebra_ii-help/write-a-polynomial-function-from-its-zeros

https://tutorial.math.lamar.edu/problems/alg/FindingZeroesOfPolynomials.aspx

Thank you for the links. I will practice solving a few problems.