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MICROSOF T Excel 2010

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VBA pour Excel 2010, 2013 et 2016 - fnac-staticcom

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VBA pour Excel 2010, 2013 et 2016 - fnac-staticcom

20 MATHEMATICS

File Name : C:\Computer Station\Class - X (Maths)/Final/Chap-2/Chap-

2(8th Nov).pmd

2

2.1 Introduction

In Class IX, you have studied polynomials in one variable and their degr ees. Recall that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y 2 - 3y + 4 is a polynomial in the variable y of degree 2, 5x 3 - 4x 2 + x - 2 is a polynomial in the variable x of degree 3 and 7u 6 42

3482uuu

is a polynomial in the variable u of degree 6. Expressions like 1 1x 2x 2 1 23xx
etc., are not polynomials. A polynomial of degree 1 is called a linear polynomial. For example, 2x - 3, 35,x
2y 2 11x , 3z + 4, 213u
, etc., are all linear polynomials. Polynomials such as 2x + 5 - x 2 , x 3 + 1, etc., are not linear polynomials. A polynomial of degree 2 is called a quadratic polynomial. The name ‘quadratic" has been derived from the word ‘quadrate", which means ‘square" 2

2,235xx

y 2 - 2, 2 23,xx
22 2

2125,5 ,4337uuvvz

are some examples of quadratic polynomials (whose coefficients are real numbers). More generally, any quadratic polynomial in x is of the form ax 2 + bx + c, where a, b, c are real numbers and a ≠ 0. A polynomial of degree 3 is called a cubic polynomial. Some examples of

POLYNOMIALS

© NCERT

not to be republished

POLYNOMIALS21

File Name : C:\Computer Station\Class - X (Maths)/Final/Chap-2/Chap-

2(8th Nov).pmd

a cubic polynomial are 2 - x 3 , x 3 3 2,x 3 - x 2 + x 3 , 3x 3 - 2x 2 + x - 1. In fact, the most general form of a cubic polynomial is ax 3 + bx 2 + cx + d, where, a, b, c, d are real numbers and a ≠ 0.

Now consider the polynomial p(x) = x

2 - 3x - 4. Then, putting x = 2 in the polynomial, we get p(2) = 2 2 - 3 × 2 - 4 = - 6. The value ‘- 6", obtained by re placing x by 2 in x 2 - 3x - 4, is the value of x 2 - 3x - 4 at x = 2. Similarly, p(0) is the value of p(x) at x = 0, which is - 4. If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k, and is denoted by p(k).

What is the value of p(x) = x

2 -3x - 4 at x = -1? We have : p(-1) = (-1) 2 -{3 × (-1)} - 4 = 0

Also, note thatp(4) = 4

2 - (3 × 4) - 4 = 0. As p(-1) = 0 and p(4) = 0, -1 and 4 are called the zeroes of the quadratic polynomial x 2 - 3x - 4. More generally, a real number k is said to be a zero of a polynomial p(x), if p(k) = 0. You have already studied in Class IX, how to find the zeroes of a linear polynomial. For example, if k is a zero of p(x) = 2x + 3, then p(k) = 0 gives us

2k + 3 = 0, i.e., k =

3 2 In general, if k is a zero of p(x) = ax + b, then p(k) = ak + b = 0, i.e., bka

So, the zero of the linear polynomial ax + b is

(Constant term)

Coefficient ofb

axquotesdbs_dbs2.pdfusesText_3