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anemone
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Prove that there are only two real numbers such that $(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=720$.
MarkFL said:Addendum to my solution:
After dividing, we found:
\(\displaystyle f(x)=x^4-14x^3+77x^2-196x+252\)
We may write this as:
\(\displaystyle \left(x^2-7x \right)^2+28\left(x^2-7x \right)+252=\left(x^2-7x+14 \right)^2+56\)
This is enough to show that this function as no real roots.
A polynomial is a mathematical expression consisting of variables and coefficients, combined using the operations of addition, subtraction, and multiplication. The highest exponent of the variable in a polynomial is called the degree.
The degree of a polynomial is determined by the highest exponent of the variable. In this case, the highest exponent is 6, so the degree of this polynomial is 6.
The polynomial (x-1)(x-2)(x-3)(x-4)(x-5)(x-6) is equal to 720 when x takes on the values of 7, 8, 9, 10, 11, and 12. This is because when these values are substituted into the polynomial, they result in a product of 720.
To solve a polynomial means to find the values of the variable(s) that make the polynomial true. In this case, we are trying to find the values of x that make the polynomial (x-1)(x-2)(x-3)(x-4)(x-5)(x-6) equal to 720.
The polynomial (x-1)(x-2)(x-3)(x-4)(x-5)(x-6) can be expanded and simplified to show that it is equal to 720. Another way to prove it is by using mathematical induction, where we can show that the polynomial is equal to 720 for a specific value of x, and then prove that it holds for all subsequent values of x.