10.POLYNOMIAL BASICS CONCEPTS

 

जब 3 से अधिक terms एक साथ आ जाएं तो उसे ही Polynomial  कहा जाता है |

Denoted by P(x

 

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1.  VARIABLE:       A letter that represent an unknown number. x,y,z,a,b,c,p,q,r..... 

                                 

 

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2.  CONSTANT:.   value or number that never changes ,

Ex.     2, 5, 0, -3, -7, 2/7, 7/9 etc.

              

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• 3.   TERM:          
•                                
• x² ,5x,  y² /3.   ,  7 Z³ / 5,   and 2 are the terms of the polynomial.  

                                       

                                     

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4.   ALGEBRAIC EXPRESSION.    

An algebraic expression  consisting one or more terms in which variables may have anything as power including  positive, negative or fractions. 

EXAMPLE:  

6x² -7x + 5

5+1/y = 5 + y-¹

3 + √x - x²....... etc

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5.   POLYNOMIAL:     

A polynomial is an algebraic expression containing one or more terms in which the  power of the variable is always  a whole numbers. 

EXAMPLES: 

3x²,

4y - 8

8y ³+  5y² + y .  etc.

                             

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6.   TERMS AND COEFFICIENTS OF A          

        POLYNOMIAL

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7.    TYPES OF A POLYNOMIAL:   

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8.    DEGREE OF A POLYNOMIAL :

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9.    ZEROES OF A POLYNOMIAL.     

the points where the polynomial becomes zero as a whole. 

For example, consider f(x) = 3x – 12. Now, put x = 4 in the polynomial, i.e., f(4) = 3×4 – 12 = 0. Thus, x = 4 is a zero of polynomial f(x) = 3x – 12.

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10.   REMAINDER THEOREM

If a polynomial p(x) is divided by the binomial x-a, the remainder obtained is p(a).

 example,

 if p(x) = x³- 4x² - 7x + 10 was divided by 

(x-2), the remainder can be determined by finding p(2).

p(x) = x³- 4x² - 7x + 10

p(2) = (2)³ - 4(2)² - 7(2) + 10

= 8-16-14 + 10 = -12

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11.   FACTOR THEOREM 

If f(x) is a polynomial of degree n greater than or equal to 1, and 'a' is any real number, then (x - a) is a factor of f(x) if f(a) = 0.

Example:    P(x) = x² - 5x + 6

Put x= 2., P(2) = 2² -5×2 + 6. = 0          

                                                      (Reminder=0)

so (x - 2) is a factor of x² - 5x + 6.

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12.   FACTORISATION OF A POLYNOMIAL

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13.   GRAPHICAL REPRESENTATIONS

The graph of P(x) depends upon its degree. A polynomial having one variable which has the largest exponent is called a degree of the polynomial.

• Constant Polynomial Function: P(x) = a = ax0
• Zero Polynomial Function: P(x) = 0; where all ai’s are zero, i = 0, 1, 2, 3, …, n.
• Linear Polynomial Function: P(x) = ax + b
• Quadratic Polynomial Function: P(x) = ax2+bx+c
• Cubic Polynomial Function: ax3+bx2+cx+d
• Quartic Polynomial Function: ax4+bx3+cx2+dx+e

  GRAPHICAL REPRESENTATIONS IS NOT MUST.)....

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14.   RELATIONSHIP BETWEEN ZEROES 

        AND COEFFICIENTS

                    

I. Linear polynomial     ( ax+b)

Zero of Polynomial  (α) =  -b/a

Quadratic polynomial      ( ax² + bx + c )

Sum of zeroes.       (α+ β).   =     -b/a

Product of zeroes   (α. β)    = c/a

Cubic polynomial    ( ax³ + bx² + cx +  d )

Sum of zeroes.   (α+ β+ γ)   = -b/a

Sum of the product of zeroes.( α.β+ βγ+γα)   = c/a

Product of zeroes (α.β. γ)     -d/a

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Polynomial Identities

1. (x + y)2= x2 + 2xy + y2
2. (x – y)2= x2 – 2xy + y2
3. x2– y2 = (x + y)(x – y)
4. (x + a)(x + b) = x2+ (a + b)x + ab
5. (x + y + z)2= x2 + y2 + c2 + 2xy + 2yz + 2zx
6. (x + y)3= x3 + y3 + 3xy (x + y)
7. (x – y)3= x3 – y3 – 3xy (x – y)
8. x3+ y3 = (x + y)(x2 – xy + y2)
9. x3– y3 = (x – y)(x2 + xy + y2)
10. x3+ y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)

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