Question

find the zeroes of the following polynomials by the factorisation method and verify the relations between the zeroes and the coefficient of the polynomial
1.
$4x {}^{2} - 3x - 1$
2.
$3x {}^{2} + 4x - 4$
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Answer 1

4x² -3x-1 =0

4x² -4x+x -1 =0

4x(x-1)+1(x-1) =0

(4x+1)(x-1) =0

x= -1/4 and x= 1

sum of roots = a+b = -1/4 +1 = -1+4 /4 = 3/4

product of roots = ab = -1/4×1 = -1/4

3x²+4x-4 =0

3x² + 6x-2x -4 =0

3x(x+2) -2(x+2 ) =0

3x-2 = 0 and x+2 = 0

x= 2/3 and x= -2

sum of roots = 2/3 -2 = 2-6/3 = -4/3

product of roots = 2/3×-2 = -4/3

Answer 2

Answer:

1) let f(x) = 4x2 - 3x - 1

Splitting the middle term

4x2 - 4x + 1x - 1

=4x(x−1)+1(x−1)

=(x−1)(4x+1)

For f(x) = 0

4x2−3x−1=0

or (x−1)(4x+1)=0

Either x−1=0⇒x=1

or 4x+1=0

⇒4x=−1

x = -1/4

Therefore the zeroes are 1 and -1/4

VERIFICATION

α = 1 , β = -1/4

a = 4

b = -3

c = -1

α+β = -b/a

1 - 1/4 = -(-3)/4

3/4 = 3/4

LHS = RHS

αβ = c/a

1 * (-1/4) = -1/4

-1/4 = -1/4

LHS = RHS

HENCE VERIFIED