If the zero of the polynomial x2-5x+k are the reciprocal of each other find the value of k
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To find the value of k in the polynomial x^2 - 5x + k, where the zeros are reciprocal of each other, we can use the following steps:
Let's assume one of the zeros as a, and since the zeros are reciprocal, the other zero would be 1/a.
1. Write out the equation using the zeros:
(x - a)(x - 1/a) = 0
2. Expand the equation:
x^2 - ax - 1/a + a^2 = 0
3. Combine like terms:
x^2 - ax + a^2 - 1/a = 0
4. The polynomial given is x^2 - 5x + k. By comparing coefficients, we can conclude that a^2 - 1/a = k.
5. To find the value of k, we need to solve the equation a^2 - 1/a = k.
From this point, we need more information or a specific value to determine the exact value of k.
Good
If the roots are p and q, then we have
pq = k
Since p = 1/q, pq = 1
So, k=1
x^2-5x+1 has roots
(5±√21)/2
I leave it to you to verify that the two values are reciprocals.