Write a polynomial whose zeroes are 3/5 and -1/2
Answers
We're going to form a quadratic polynomial (second degree polynomial).
Given that the zeroes of the polynomial that has to be formed should be 3/5 and -1/2.
Since 3/5 and -1/2 are zeroes of the polynomial, the factors can be (x - 3/5) and (x + 1/2) respectively.
By taking these as the factors we can make a polynomial:
(x - 3/5)(x + 1/2) = x² - 1/10 x + 3/10
We can also make polynomials with same zeroes by removing the fractional form of the coefficients of x and x⁰.
Consider the factor (x - 3/5). We can multiply 5 with this factor to get (5x - 3).
Consider the factor (x + 1/2). We can multiply 2 with this factor to get (2x + 1).
Since 5 and 2 are multiplied with the factors, the resultant polynomial will be 5 × 2 = 10 times the polynomial formed first.
Hence, the resultant polynomial will be,
10(x² - 1/10 x + 3/10) = 10x² - x + 3
Using 10x² - x + 3 instead of x² - 1/10 x + 3/10 may be easier, right?!
We can make more polynomials by multiplying any constant or any other polynomial with this one. On multiplying any other polynomial, the no. of zeroes will increase.
Answer:
3/5 and -1/2 are the zeroes of a p(x)
Sum of zeroes = 3/5 + (-1/2)
= 6-5/10
=1/10
Product of zeroes = 3/5 × (-1/2)
= -3/10
Then,
p(x)= k{x2 - (sum of zeroes)x+(product of zeroes)}
let k= 10
= 10{x2 - (1/10)x + (-3/10)}
= 10{x2-1/10x - 3/10}
= 10x2-x -3
therefore, 10x2 - x -3 is the required polynomial
Hope it helps