Find a quadratic polynomial whose zeroes are 5-3√2 and 5+3√2. 2
Answers
Answer :
x² - 10x + 7
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 .
★ The general form of a quadratic polynomial is given as ; ax² + bx + c .
★ 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 a quadratic polynomial , then that quadratic polynomial is given as : k•[ x² - (α + ß)x + αß ] , k ≠ 0.
Solution :
Here ,
We need to find a quadratic polynomial whose zeros are (5 - 3√2) and (5 + 3√2) .
Let α = (5 - 3√2) and ß = (5 + 3√2) .
Thus ,
The sum of zeros of the required quadratic polynomial will be ;
=> α + ß = (5 - 3√2) + (5 + 3√2)
=> α + ß = 10
Also ,
The product of zeros of the required quadratic polynomial will be ;
=> αß = (5 - 3√2)(5 + 3√2)
=> αß = 5² - (3√2)²
=> αß = 25 - 18
=> αß = 7
Also ,
We know that , if α and ß are the zeros of a quadratic polynomial , then that quadratic polynomial is given as : k•[ x² - (α + ß)x + αß ] , k ≠ 0.
Thus ,
Required quadratic polynomial will be ;
k•[ x² - 10x + 7 ] , k ≠ 0
If k = 1 , then the quadratic polynomial will be ; x² - 10x + 7 .