If alpha and beta are the zeroes of the quadratic polynomial f(x) =2x^2-6x+7 find a polynomial whose zeroes are 2 alpha+ 3 beta and 3 alpha+ 2 beta
Answers
Question:
If @ and ß are the zeroes of the quadratic polynomial f(x) =2x² - 6x + 7 . Find a polynomial whose zeroes are 2@ + 3ß and 3@ + 2ß .
Answer:
x² - 15x + 115/2
Note:
• The general for of a quadratic polynomial is given as : ax² + bx + c .
• Zeros of a polynomial are the possible values of unknown (variable) for which the polynomial becomes zero.
• If A and B are the zeros of quadratic polynomial ax² + bx + c then ;
Sum of zeros , (A+B) = -b/a
Product of zeros , (A•B) = c/a
• If A and B are the zeros of any quadratic polynomial then that quadratic polynomial is given as : x² - (A+B)x + A•B
Solution:
The given quadratic polynomial is :
f(x) = 2x² - 6x + 7.
Clearly, we have ;
a = 2
b = -6
c = 7
Also,
It is given that , @ and ß are the zeros of the given polynomial f(x) , thus ;
=> Sum of zeros = -b/a
=> @ + ß = -(-6)/2
=> @ + ß = 6/2
=> @ + ß = 3 -------(1)
Also,
=> Product of zeros = c/a
=> @•ß = 7/2 --------(2)
Now,
Let A and B be the zeros of required quadratic polynomial.
Thus,
A = 2@ + 3ß
B = 3@ + 2ß
Now,
Sum of zeros of required quadratic polynomial will be given as ;
=> A + B = 2@ + 3ß + 3@ + 2ß
=> A + B = 5@ + 5ß
=> A + B = 5(@ + ß)
=> A + B = 5•3 {using eq-(1)}
=> A + B = 15
Also,
Product of zeros of required quadratic polynomial will be given as ;
=> A•B = (2@ + 3ß)•(3@ + 2ß)
=> A•B = 6@² + 4@•ß + 9@•ß + 6ß²
=> A•B = 6(@² + ß²) + 13@•ß
=> A•B = 6[(@ + ß)² - 2@•ß] + 13@•ß
=> A•B = 6(@ + ß)² - 12@•ß + 13@•ß
=> A•B = 6(@ + ß)² + @•ß
=> A•B = 6•3² + 7/2 {using eq-(1) and eq-(2)}
=> A•B = 54 + 7/2
=> A•B = (108 + 7)/2
=> A•B = 115/2
Now,
The required quadratic polynomial will be given as ; x² - (A + B)x + A•B
ie ; x² - 15x + 115/2
Hence,
The required quadratic polynomial is :
x² - 15x + 115/2 .
Answer:
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