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Polynomial Class 10th

Degree of a polynomial – The highest power of the variable x in a polynomial is called the degree of the polynomial.

Example – 5x⁴ + 3x² + 2x + 3

Linear polynomial – A polynomial in which the maximum power of the variable is 1 is called a linear polynomial.

Example – 5x – 4, 2t-3

Quadratic polynomial – A polynomial in which the maximum power of the variable is 2 is called a quadratic polynomial.

Example – 3x² + 4x – 3, 2x² + 3

Cubic polynomial – A polynomial in which the maximum power of the variable is 3 is called a cubic polynomial.

Example – 4x³ + 3x² + 7

Standard equation of a polynomial –

Exercise – 1.2

Que. 1. Find the zeroes of the following quadratic polynomials and check the truth of the relationship between the zeroes and the coefficients –

1. x² -2x -8

Sol. x² -4x +2x -8

x (x-4) +2 (x-4)

(x-4) (x+2)

x-4 =0                                  x+2 =0

x =4                                      x = -2

Standard equation

ax² + bx + c

x² -2x -8

a = 1,              b = -2,                     c = -8

Relationship between zeroes and coefficients-

Sum of zeroes

4 -2 = 2

2 = 2

Product of zeroes

-8 = -8

2. 4s² -4s +1

Sol. 4s² -2s -2s +1

2s (2s -1) -1 (2s -1)

2s -1 = 0                                2s -1 = 0

2s = 1                                      2s = 1

Standard equation

ax² + bx + c

4s² -4s +1

a = 4,                              b = -4,                    c = 1

Relationship between zeroes and coefficients –

Sum of zeroes

Product of zeroes

3. 6x² -3 -7x

Sol. 6x² -7x -3

6x² -9x +2x -3

3x (2x -3) +1 (2x -3)

(2x -3) (3x +1)

2x -3 = 0                      3x +1 = 0            

2x = 3                            3x = 1

Sum of zeroes

Product of zeroes

4. 4u² + 8u

Sol. 4u² + 8u

4u (u + 2)

4u = 0                     u + 2 = 0

u = 0/4                          u = -2

u = 0

Sum of zeroes

Product of zeroes

5. t² – 15

Sum of zeroes  

Product of zeroes

6. 3x² -x -4

Sol. 3x² -4x +3x -4

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

(3x -4) (x +1)

3x -4 = 0                    x +1 = 0

3x = 4                          x = -1

Sum of zeroes

Product of zeroes

Que. 2. Find a quadratic polynomial whose sum and product of zeros are the given numbers, respectively.

Sol. x² – (Sum of zeroes) x + Product of zeroes

Sol. x² – (Sum of zeroes) x + Product of zeroes

Sol. x² – (Sum of zeroes) x + Product of zeroes

4. 1,1

Sol. x² – (Sum of zeroes) x + Product of zeroes

x² –1x +1

x² –x +1

6. 4,1

Sol. x² – (Sum of zeroes) x + Product of zeroes

x² –4x +1