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'Roots' aren't like prime factors; repeating a root isn't meaningful. In prime factorization, 2 is the only distinct prime factor of 2 and 32, but since 2 goes into 2 once and 32 five times, 32 have five prime factors. With roots, however, either a given value of X does correctly balance the equation, or it doesn't; it's a meaningless distinction to say that -1 can 'solve the equation twice.' Thus, since there are exactly two x values that solve the equation, 1 and -1, we say there are exactly two roots, even if as a 4th degree equation each of those roots could be derived two ways.
How many roots does the equation x^4 - 2x^2 +1=0 have?
a) 0 b) 1 c) 2 d) 3 e) 4
As per the solution given, x = 1 or x = -1.
My question is: We are asked how many roots are there for this equation not how many different roots. So, total no. of roots should be 4.
Experts please comment.
technically, number of the roots of polynomial is max power in polynomial. so that way answer must be E. i.e. number of roots of given equation is 4. out 4, 2 are same. here we are not concerned whether roots are repeated or not. we are asked to find number of roots. ..please comment. thanks
@lalab Thanks for your response. I am trying to find problem in my approach.
ax^2+bx+c=0. This is a polynomial of degree 2, where D=b^2-4ac
For a polynomial of degree 4, i.e. maximum power of x, falls in a different category: ax^4+bx^2+c=0. This is a polynomial of degree 4, for which D = b^2-4ac may not hold true.
The question is asking about: x^4 - 2x^2 +1=0. The maximum power of x is 4. You just don't know what is the discriminant. The discriminant formula you used is only for quadratic equations(x with highest power of 2).
For a quadratic equation, your approach would be correct.