Step-by-Step Solution

Step 1: Remove Fractions

To simplify the polynomial, we first remove fractions by multiplying the expression suitably:

So,

1/4 [ 8x² + 14x + 3 ]

Step 2: Split the Middle Term

Split the middle term 14x as 12x + 2x:

1/4 [ 8x² + 12x + 2x + 3 ]

Step 3: Factorisation

Group the terms:

1/4 [ 4x(2x + 3) + 1(2x + 3) ]

Taking (2x + 3) as common factor:

p(x) = 1/4 (2x + 3)(4x + 1)

Finding the Zeroes

To find the zeroes, equate p(x) = 0:

1/4 (2x + 3)(4x + 1) = 0

So,

• 2x + 3 = 0 → x = -3/2
• 4x + 1 = 0 → x = -1/4

Therefore, the zeroes are:

α = -3/2, β = -1/4

Verification of Relations Between Zeroes and Coefficients

Sum of Zeroes (α + β)

α + β = -3/2 + (-1/4)

= -6/4 – 1/4

= -7/4

According to the formula:

α + β = -b/a

Hence, the relation is verified.

Product of Zeroes (αβ)

αβ = (-3/2)(-1/4)

= 3/8

According to the formula:

αβ = c/a

Hence, the relation between zeroes and coefficients is verified.

Video Explanation

Why This Question Is Important for Class 10 CBSE Exams

NCERT Exemplar questions like this one are frequently tested because they check:

• Concept clarity of polynomials
• Correct use of factorisation
• Ability to verify results using formulas
• Step-by-step presentation as per CBSE marking scheme

Students who practise such questions regularly make fewer mistakes in board exams and score better.

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