What is minimum value of |a+bw+cw^2|? whrere w is cube root of unity?

In summary, the minimum value of |a+bw+cw^2| can be calculated using the quadratic formula and the fact that w is the cube root of unity. The minimum value occurs when w = 1, and is equal to |a+b+c|. However, this value can vary based on the coefficients of the equation and can also be negative. This minimum value can be used in various mathematical applications, such as finding the minimum value of a function or determining the minimum distance between a point and a curve. It can also provide insight into the behavior of mathematical equations.
  • #1
vkash
318
1
a,b,c are integers not all equal and w is the cube root of unity then minimum value of |a+bw+cw2|(w is not equals one).

My answer
|a+bw+cw2|<=|a|+|bw|+|cw2|
|a|+|bw|+|cw2|=a+b+c.
so at lest one value of |a+bw+cw2| will smaller than the minimum value of a+b+c. for integers this minimum value is smallest integers you can think of.
But that's wrong.?why?
 
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  • #2
Is |w2| = 1?
 

Related to What is minimum value of |a+bw+cw^2|? whrere w is cube root of unity?

1. What is the minimum value of |a+bw+cw^2|?

The minimum value of |a+bw+cw^2| depends on the values of a, b, and c. It can be calculated using the quadratic formula and the fact that w is the cube root of unity, which means that w^3 = 1. The minimum value occurs when w = 1, and the minimum value is equal to |a+b+c|.

2. How do you calculate the minimum value of |a+bw+cw^2|?

To calculate the minimum value of |a+bw+cw^2|, you can use the quadratic formula and plug in the values of a, b, and c. You also need to use the fact that w is the cube root of unity, which means that w^3 = 1. The minimum value occurs when w = 1, and the minimum value is equal to |a+b+c|.

3. Is the minimum value of |a+bw+cw^2| always the same?

No, the minimum value of |a+bw+cw^2| is not always the same. It depends on the values of a, b, and c. The minimum value can vary based on the coefficients of the equation.

4. Can the minimum value of |a+bw+cw^2| be negative?

Yes, the minimum value of |a+bw+cw^2| can be negative. This can occur if the values of a, b, and c are such that the minimum value, |a+b+c|, is negative. In this case, the absolute value of the expression becomes the negative minimum value.

5. How can the minimum value of |a+bw+cw^2| be used in mathematics?

The minimum value of |a+bw+cw^2| can be used in various mathematical applications, such as finding the minimum value of a function or determining the minimum distance between a point and a curve. It can also be used in optimization problems to find the minimum possible value of an expression. Additionally, the minimum value can provide insight into the behavior of a mathematical equation or function.

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