Polynomial Challenge: Find Real Solutions

In summary, a polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication operations. Finding real solutions means finding the values of the variables that make the equation true, and these solutions can be determined through techniques such as factoring, the quadratic formula, or the rational roots theorem. A polynomial can have multiple real solutions, and finding these solutions is important in solving real-world problems and understanding physical and scientific phenomena.
  • #1
anemone
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Find the number of distinct real solutions of the equation

$(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.
 
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  • #2
anemone said:
Find the number of distinct real solutions of the equation

$(x − 1)(x − 3)(x − 5) · · · (x − 2017) = (x − 2)(x − 4)(x − 6) · · · (x − 2016)$.

the above is same as
$P(x) = (x - 1)(x - 3)(x - 5) \cdots (x - 2017) - (x - 2)(x - 4)(x - 6) \cdots (x - 2016)= 0$
This is a polynomial of degree 1009.
this has product of 1009 terms in the 1st term
now let us compute P(2n) for n = 1 to 1008
this is +ve for n odd( as in 1st term there are n positive and 1009-n -ve terms
and 2nd term is zero) and -ve for n even. and $P(-\infty) < 0$ and $P(\infty) > 0$ so sign changes 1009 times
so 1009 real roots.
 

FAQ: Polynomial Challenge: Find Real Solutions

What is a polynomial?

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication operations. It may also include exponents, but the variables must have whole number exponents. Examples of polynomials include 2x^2 + 3x - 5 and 4y^3 + 9y^2 + 2y + 6.

What does it mean to find real solutions?

When solving a polynomial, finding real solutions means finding the values of the variables that make the equation true. These values are called roots or solutions. Real solutions are those that result in real numbers when substituted into the equation.

How do you solve a polynomial?

To solve a polynomial, you can use techniques such as factoring, the quadratic formula, or the rational roots theorem. These methods involve manipulating the equation to isolate the variable and determine its value. It is also helpful to graph the polynomial to visually determine the solutions.

Can a polynomial have more than one real solution?

Yes, a polynomial can have multiple real solutions. For example, the polynomial x^2 - 4 has two real solutions, x = 2 and x = -2. However, some polynomials may only have one real solution or no real solutions at all.

Why is finding real solutions important?

Finding real solutions is important because it allows us to solve real-world problems and make predictions based on mathematical models. Many physical and scientific phenomena can be represented by polynomials, so finding real solutions is crucial in understanding and analyzing these phenomena.

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