Form the quadratic polynomials whose zeroes are : a)3+√2. b)-√2 and √2. c)1÷3 and 1÷4. d)-5 and -3 e) 3 and 1÷5. f)1÷a. , 1÷b. pls dont give old answers and answer fast pls
Answers
Answer:
a). x² - (3 + √2)x + 3√2
b). x² - 2
c). (1/12)•(12x² - 7x + 1) OR (12x² - 7x + 1)
d). x² + 8x + 15
e). (1/5)•(5x² - 16x + 3) OR (5x² - 16x + 3)
f). (1/ab)•[abx² - (a + b)x + 1]
OR abx² - (a + b)x + 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 it is given by ;
x² - (α + ß)x + αß
★ If α and ß are the zeros of the quadratic polynomial ax² + bx + c , then they (α and ß) are also the zeros of the quadratic polynomial k(ax² + bx + c) , k≠0.
Solution:
We need to form quadratic polynomial using the given zeros .
Also,
Let α and ß be the zeros in every case .
Thus, Here we go ↓
(a) 3 and √2
α = 3
ß = √2
Sum of zeros , (α + ß) = 3 + √2
Product of zeros , αß = 3√2
Thus,
The required quadratic polynomial will be given as ; x² - (α + ß)x + αß
ie ; x² - (3 + √2)x + 3√2
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°
(b) -√2 and √2
α = -√2
ß = √2
Sum of zeros , (α + ß) = -√2 + √2 = 0
Product of zeros , αß = -√2•√2 = -2
Thus,
The required quadratic polynomial will be given as ; x² - (α + ß)x + αß
ie ; x² - 0•x + (-2)
ie; x² - 2
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°
(c) 1/3 and 1/4
α = 1/3
ß = 1/4
Sum of zeros , (α + ß) = 1/3 + 1/4
= (4 + 3)/12
= 7/12
Product of zeros , αß = (1/3)•(1/4) = 1/12
Thus,
The required quadratic polynomial will be given as ; x² - (α + ß)x + αß
ie ; x² - (7/12)x + 1/12
ie ; x² - 7x/12 + 1/12
ie; (1/12)•(12x² - 7x + 1)
Another quadratic polynomial may be ;
12x² - 7x + 1
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°
(d) -5 and -3
α = -5
ß = -3
Sum of zeros , (α + ß) = -3-5 = -8
Product of zeros , αß = (-3)•(-5) = 15
Thus,
The required quadratic polynomial will be given as ; x² - (α + ß)x + αß
ie ; x² - (-8)x + 15
ie; x² + 8x + 15
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°
(e) 3 and 1/5
α = 3
ß = 1/5
Sum of zeros , (α + ß) = 3 + 1/5
= (15 + 1)/5
= 16/5
Product of zeros , αß = 3•(1/5) = 3/5
Thus,
The required quadratic polynomial will be given as ; x² - (α + ß)x + αß
ie ; x² - (16/5)x + 3/5
ie ; x² - 16x/5 + 3/5
ie ; (1/5)•(5x² - 16x + 3)
Another quadratic polynomial may be ;
x² - 16x + 3
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°
(f) 1/a + 1/b
α = 1/a
ß = 1/b
Sum of zeros , (α + ß) = 1/a + 1/b
= (b + a)/ab
= (a + b)/ab
Product of zeros , αß = (1/a)•(1/b) = 1/ab
Thus,
The required quadratic polynomial will be given as ; x² - (α + ß)x + αß
ie ; x² - [ (a + b)/ab ]x + 1/ab
ie ; x² - (a + b)x/ab + 1/ab
ie ; (1/ab)•[abx² - (a + b)x + 1]
Another quadratic polynomial may be ;
abx² - (a + b)x + 1
°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°°