Question

Find the zeroes of the quad valtc Polynomial 12x²-4x+3x-1
and verify the relationship between the zeroes and
Coefficients​

Answer 1

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☆ Given :

Polynomial = 12x²-4x+3x-1 i.e.12x² -x -1

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☆ To Calculate :

Zeros of the polynomial 12x²-4x+3x-1.

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☆ Also To Verify :

Relationship between Zeros And Coefficient.

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¤ Relationship between

Zeros And Coefficients :

If a Polynomial is of the Form ax² + bx + c.

and it's zeros are α and β.

Then,

α + β = -b/a

and

αβ = c/a

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☆ Solution :

《♤ {Calculating Zeros}♤》

12x²-4x+3x-1

= 4x(3x - 1) + 1(3x - 1 )

= (3x - 1)(4x+1)

So,

Zeros of the given polynomial are 1/3 and -1/4.

●》α= 1/3 and β = -1/4

《♤{Verifying Releationship}♤》

$\sf\alpha + \beta = \frac{1}{3} + ( \frac{ - 1}{4} ) \\ \\ = \frac{1}{3} - \frac{1}{4} \\ \\ = \frac{1}{12} = \bf- \frac{b}{a} \: \: \: \{verified \}$

$\sf\alpha \beta = \frac{1}{3} \times ( - \frac{1}{4} ) \\ \\ = - \frac{1}{12} = \bf \frac{c}{a} \: \: \: \: \{verified \}$

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Answer 2

Given :-

Quadratic polynomial = 12x² - 4x + 3x - 1

To Find :-

Zeroes and verify them

Solution :-

12x² - 4x + 3x - 1 = 0

12x² - x - 1 = 0

12x² - (4x - 3x) - 1 = 0

12x² - 4x + 3x - 1 = 0

4x(3x - 1) + 1(3x - 1) = 0

(3x - 1)(4x + 1) = 0

Finding zeroes

3x - 1 = 0

3x = 0 + 1

3x = 1

x = 1/3

Or

4x + 1 = 0

4x = 0 - 1

4x = -1

x = -1/4

So, the zeroes will be 1/3 and -1/4

Now

Verification

Sum of zeroes = α + β = -b/a

1/3 + (-1)/4 = -(-1/12)

1/3 - 1/4 = 1/12

4 - 3/12 = 1/12

1/12 = 1/12

Product of zeroes = α × β = c/a

1/3 × -1/4 = -1/12

-1/12 =  -1/12