anemone said:Prove that $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$ is never equal to 33.
kaliprasad said:First let us factor $m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$
$m^5+3m^4n-5m^3n^2-15m^2n^3+4mn^4+12n^5$
= $m^4(m+3n) - 5m^2n^2(m+3n) + 4n^4(m+3n)$
= $(m+3n)(m^4-5m^2n^2+4n^4)$
= $(m+3n)(m^2-4n^2)(m^2-n^2)$
= $(m+3n)(m-2n)(m+2n)(m-n)(m+n)$
= $(m-2n)(m-n)(m+n)(m+2n)(m+3n)$
for n = 0 the value is $m^5$ and 33 is not a $5^{th}$ power
for n not zero above 5 values are differnt and 33 is product of atmost 4 mumbers $( (- 1) * 1 * (-3) * 11$ and hence product
cannot be 33
The first step in proving that the expression is not equal to 33 is to set it equal to 33 and then simplify it using algebraic techniques. If you are able to reach a point where the left side does not equal the right side, then you have successfully proven that the expression is not equal to 33.
Yes, you can choose specific values for m and n and substitute them into the expression. If the resulting simplified expression is not equal to 33, then you have proven that the expression is not equal to 33 for those particular values of m and n.
There is not one specific property or principle that you must use to prove this expression is not equal to 33. You may need to use a combination of algebraic techniques, such as factoring, distributing, or simplifying, to reach a point where the left side does not equal the right side.
Yes, you can use a proof by contradiction to prove that the expression is not equal to 33. In this type of proof, you assume that the expression is equal to 33 and then use logical reasoning and mathematical operations to reach a contradiction, thus proving that the expression cannot be equal to 33.
One common mistake to avoid is assuming that the expression is not equal to 33 without fully simplifying and evaluating both sides of the equation. It is important to follow all necessary steps and not skip any in order to reach a valid conclusion. Additionally, be careful when manipulating terms and variables, as a small mistake can lead to an incorrect solution.