maths pls solve i am from Hyderabad
Answers
Step-by-step explanation:
Solution :-
1)
Given zeroes are -3 and 2
Let α = -3
Let β = 2
We know that
The quadratic polynomial whose zeores are α and β is K[x²-(α +β)x + α β]
On Substituting these values in the above formula
=> K[x²-(-3+2)x+(-3)(2)]
=> K[x²-(-1)x+(-6)]
=> K[x²+1x-6]
=>K[x²+x-6]
If K = 1 then the required quadratic polynomial is x²+x-6.
2)
Given bi quardratic polynomial is
2t⁴+3t³-2t²-9t-12
Given another Polynomial = t²-3
t²-3 ) 2t⁴+3t³-2t²-9t-12 ( 2t²+3t+4
2t⁴ -6t²
(-) (+)
_____________
0 +3 t³ + 4t² -9t
3 t³ -9t
(-) (+)
______________
0 +4 t² +0 -12
4 t² -12
(-) (+)
________________
0
________________
So we get remainder as 0
If any polynomial divides the other Polynomial completely then the polynomial is a factor of the other polynomial
So, t²-3 is a factor of the given bi Quadratic Polynomial.
3)
Given cubic polynomial P(x) = x³-3x²+x+1
On Comparing this with the standard cubic Polynomial ax³+bx²+cx+d
a = 1
b= -3
c=1
d = 1
Given zeroes are (a-b) , a and (a+b)
We know that
Sum of the zeroes = -b/a
=> a+b+a+a-b =- (-3)/1
=> 3a = 3
=> a = 3/3
=> a = 1
and
Product of the zeores = -d/a
=> (a-b)(a)(a+b) = -1/1
=> (1-b)(1)(1-b) = -1
=> (1-b)(1+b) = -1
=> 1-b² = -1
=> 1+1 = b²
=> b² = 2
=> b = ±√2
The value of a and b are 1 and ±√2 respectively.
4)
Given Quadratic Polynomial = x²+7x+10
Finding the zeores :-
=> x²+7x+10
=> x²+2x+5x+10
=> x(x+2)+5(x+2)
=> (x+2)(x+5)
To get the zeores we write as
=> (x+2)(x+5) = 0
=> x+2 = 0 or x+5 = 0
=> x = -2 and x = -5
Relationship between the zeroes and the coefficients:-
On Comparing this with the standard quadratic Polynomial ax²+bx+c
a = 1
b= 7
c=10
and
Let α = -5
Let β = -2
Sum of the zeroes = α + β
=> -5-2
= -7
and
-b/a = -(-7)/1 = 7
Sum of the zeroes = -b/a
Product of the zeroes = α β
=> (-5)(-2)
=> 10
and
c/a = 10/1 = 10
Product of the zeroes= c/a
Verified the given relations in the given problem.
Used formulae:-
1. The quadratic polynomial whose zeores are α and β is K[x²-(α +β)x + α β]
2.The standard quadratic Polynomial is ax²+bx+c
3.Sum of the zeroes = -b/a
4.Product of the zeroes= c/a
5.The standard cubic Polynomial is ax³+bx²+cx+d
6. Sum of the zeroes = -b/a
7..Product of the zeroes= -d/a
8. If any polynomial divides the other Polynomial completely then the polynomial is a factor of other polynomial.