Question

Find a quadratic polynomial sum of whose zeros is 8 and their product is 12. Hence find the
zeros of the polynomial and verify the relationship between zeros and their coefficient
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Answer 1

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➽ Given :-

For a Quadratic Polynomial

   

Sum of Zeros = 8

Product of Zeros = 12

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➽ To Find :-

The Quadratic Polynomial.

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➽ Key Point :-

If sum and product of zeros of any quadratic polynomial are s and p respectively,

Then,

The quadratic polynomial is given by :-

$\bf  {x}^{2}  - s \: x + p$

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➽ Solution :-

Here,

Sum = s = 8

and

Product = p = 12

So,

Required Polynomial should be

$\bf{x}^{2}  - 8x +  12$

《Findind Zeros》

$\sf {x}^{2} - 8x + 12 \\ \\ \sf {x}^{2} - 6x - 2x + 12 \\ \\ \sf x(x - 6) - 2(x - 6) \\ \\ \sf (x - 6)(x - 2)$

So, Zeros are

α = 6

and

β = 2.

《Verifying Relationship》

$\sf Sum = \alpha + \beta =8\\\sf = - (\frac { - 8}{1} )=\frac{-b}{a}\: \: \: \bf \{verifed \}$

$\sf \: Product = \alpha \beta\\\sf = 6 \times 2 = 12 = \frac{12}{1}=\frac{c}{a} \: \: \: \bf \{verified \}$

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Answer 2

Given :-

Sum = 8

Product = 12

To Find :

Quadratic polynomial

Solution :-

We know that

Standard form of quadratic polynomial = x² - (α + β)x + αβ

x² - (8)x + 12

x² - 8 × x + 12

x² - 8x + 12

Verification :-

x² - 8x + 12 = 0

x² - (6x + 2x) + 12 = 0

x² - 6x - 2x + 12 = 0

x(x - 6) - 2(x - 6) = 0

(x - 6)(x - 2) = 0

Either

x = 6

or

x = 2

Sum of zeros = -b/a

6 + 2 = -(-8)/1

8 = 8/1

8 = 8

Product of zeroes = c/a

6 × 2 = 12/1

12 = 12