if 1,-1 and 2 are zeroes of a polynomial p(x)=x3- ax2 -bx+2 find the values of a and b
plz answer this question as soon as possible this is important
who will give answer correctly I will follow them give thanks mark as brainliest and rate you
unsual answers will be reported
plz do it fast
Answers
Answer :
a = 2, b = 1
Step-by-step explanation :
Can be solved in two ways,
- substituting the values of zeroes in place of x
- from the relationship between zeroes and coefficients
_______________________
Method - 1 :
Given polynomial,
p(x) = x³ - ax² - bx + 2
>>> 1,-1 and 2 are zeroes of a polynomial
So, when we substitute the above values in place of x ; the result is zero.
- Put x = 1,
p(1) = 0
1³ - a(1)² - b(1) + 2 = 0
1 - a - b + 2 = 0
3 - a - b = 0
- Put x = -1,
p(-1) = 0
(-1)³ - a(-1)² - b(-1) + 2 = 0
-1 - a(1) + b + 2 = 0
-1 - a + b + 2 = 0
- a + b + 1 = 0
- Put x = 2,
p(2) = 0
2³ - a(2)² - b(2) + 2 = 0
8 - a(4) - 2b + 2 = 0
8 - 4a - 2b + 2 = 0
4a + 2b = 10
2(2a+b) = 2(5)
______________________
We got three equations,
a + b = 3 --(1)
b - a = -1 --(2)
2a + b = 5 --(3)
Equation (1)
a + b = 3
a = 3 - b
substitute the value of a in equation (2)
Equation (2)
b - a = -1
b - (3-b) = -1
b - 3 + b = -1
2b = 3 - 1
2b = 2
b = 2/2
b = 1
=> a = 3 - b
= 3 - 1
= 2
Substitute the values of a and b in equation (3) to VERIFY
>> 2a + b
2(2) + 1
4 + 1
5
Therefore, a = 2, b = 1
____________________
Method - 2 :
For a cubic polynomial of the form,
ax³ + bx² + cx + d
Let α,β,γ are the zeroes
RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS :
➤ α + β + γ = -b/a
➤ αβ + βγ + αγ = c/a
➤ αβγ = -d/a
Given polynomial,
x³ - ax² - bx + 2
and its zeroes : 1,-1,2
From the above relation,
=> 1 + (-1) + 2 = -(-a)/1
1 - 1 + 2 = a
a = 2
=> (1)(-1) + (-1)(2) + (2)(1) = -b/1
-1 - 2 + 2 = -b
-b = -1
b = 1
∴ a = 2, b = 1