If alpha and beta are the zeroes of the polynomial 3x2-2x-7, then find a quadratic polynomial whose zeroes are a -2 and b-2
Answers
Answer:
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Answer:
x² + 10x/3 + 1/3 OR 3x² + 10x + 1
Note:
★ The possible values of the variable for which the polynomial becomes zero are called its zeros .
★ A quadratic polynomial can have atmost two zeros .
★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then ;
• Sum of zeros , (α + ß) = -b/a
• Product of zeros , (αß) = c/a
★ If α and ß are the zeros of any quadratic polynomial , then that quadratic polynomial is given as : k [ x² - (α + ß)x + αß ] , k ≠ 0
Solution:
Here,
The given quadratic polynomial is :
3x² - 2x - 7 .
On comparing the given quadratic polynomial with the general form ax² + bx + c ,
We have ;
a = 3 , b = -2 , c = 7
Also,
It is given that , α and ß are the zeros of the given quadratic polynomial .
Thus,
Sum of the zeros of the given quadratic polynomial will be given as ;
=> α + ß = -b/a
=> α + ß = -(-2)/3 = 2/3
Also,
Product of the zeros of the given quadratic polynomial will be given as ;
=> αß = c/a
=> αß = -7/3
Now,
It is given that , (α - 2) and (ß - 2) are the zeros of the required quadratic polynomial .
Thus,
Sum of the zeros of the required quadratic polynomial will be given as ;
(α - 2) + (ß - 2) = α + ß - 2 - 2
= (α + ß) - 4
= 2/3 - 4
= (2 - 12)/3
= -10/3
Also ,
Product of the zeros of the required quadratic polynomial will be given as ;
(α - 2)(ß - 2) = αß - 2α - 2ß + 4
= αß - 2(α + ß) + 4
= -7/3 - 2×(2/3) + 4
= -7/3 - 4/3 + 4
= (-7 - 4 + 12)/3
= 1/3
Thus,
The required quadratic polynomial will be given as : x² - [ (α - 2) + (ß - 2) ]x + (α - 2)(ß - 2)
ie ; x² - (-10/3)x + 1/3
ie ; x² + 10x/3 + 1/3
Also,
=> x² + 10x/3 + 1/3
=> (3x² + 10x + 1)/3
=> (1/3)•(3x² + 10x + 1)
Thus,
Another quadratic polynomial may be ;
3x² + 10x + 1