For $2 x^{2} + 7 x + 5$ , the sum of the zeroes and the product of the zeroes are respectively ____.

Class 10PolynomialsAlgebra — Polynomials

Step-by-Step Solution

Sum =

- \frac{b}{a} = - \frac{7}{2}

; product =

\frac{c}{a} = \frac{5}{2}

. Check by factoring:

2 x^{2} + 7 x + 5 = (2 x + 5) (x + 1)

, zeroes

- \frac{5}{2}

and −1, whose sum is

- \frac{7}{2}

and product

\frac{5}{2}

. The second option swaps sum and product; the first drops the minus on the sum.

Answer: D)

- \frac{7}{2}

and

\frac{5}{2}

If α and β are zeros of

a x^{2} + b x + c

, then:

**α + β = −

b

/a** (sum of zeros).
**α · β =

c

/a** (product of zeros).

**Why**:

a x^{2} + b x + c = a (x

− α)

(x

− β) =

a

[

x^{2}

− (α+β)

x

+ αβ].
Matching coefficients with

x^{2} + (b / a) x + (c / a)

: coefficient of

x

→ −(α+β) =

b

/a → α+β = −

b

/a, and constant: αβ =

c

/a.

**Reverse use**: To form a quadratic with given zeros α, β, write

k

[

x^{2}

− (α+β)

x

+ αβ]
for any non-zero constant

k

(usually

k = 1

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