Question 14
Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials
(i) 4x2−3x−1
Solution:
To find the zeroes of the polynomial 4x2−3x−1 using the factorization method, we will follow these steps:
Step 1: Write the polynomial equation
We start with the polynomial:
4x2−3x−1=0
Step 2: Rewrite the middle term
To factor the polynomial, we need to express −3x as a sum of two terms whose coefficients multiply to give the product of the coefficient of x2 (which is 4) and the constant term (which is −1). Thus, we need two numbers that multiply to 4×(−1)=−4 and add up to −3.
The numbers that satisfy this condition are −4 and 1. Therefore, we can rewrite the polynomial as:
4x2−4x+x−1=0
Step 3: Factor by grouping
Now we will group the terms:
(4x2−4x)+(x−1)=0
Factoring out the common terms in each group:
4x(x−1)+1(x−1)=0
Now, we can factor out (x−1):
(4x+1)(x−1)=0
Step 4: Set each factor to zero
Now we will set each factor equal to zero:
1. 4x+1=0
2. x−1=0
Step 5: Solve for x
For the first equation:
4x+1=0⟹4x=−1⟹x=−1/4
For the second equation:
x−1=0⟹x=1
Step 6: State the zeroes
Thus, the zeroes of the polynomial 4x2−3x−1 are:
x=−1/4 and x=1
Step 7: Verify the relations between the zeroes and coefficients
Let α=−1/4 and β=1.
1. Sum of the zeroes:
α+β=−1/4+1=−1/4+4/4=3/4
According to the relation, −b/a:
−[−3]/4=3/4
2. Product of the zeroes:
α⋅β=−1/4⋅1=−1/4
According to the relation, ca:
−1/4=−1/4
Both relations hold true, confirming that our zeroes are correct.
Final Answer
The zeroes of the polynomial 4x2−3x−1 are:
−1/4 and 1