NCERT Class 9 Mathematics
POLYNOMIALS 33 Solution : (i) p(x) = 5x^2 – 3x + 7 The value of the polynomial p(x) at x = 1 is given by p(1) = 5(1)^2 – 3(1) + ...
34 MATHEMATICS Now, 2 x + 1 = 0 gives us x = 1 – 2 So, 1 – 2 is a zero of the polynomial 2x + 1. Now, if p(x) = ax + b, a ✂ 0, i ...
POLYNOMIALS 35 Verify whether the following are zeroes of the polynomial, indicated against them. (i) p(x) = 3x + 1,x = 1 – 3 ...
36 MATHEMATICS Consider another pair of polynomials 3x^2 + x + 1 and x. Here, (3x^2 + x + 1) ÷ x =(3x^2 ÷ x) + (x ÷ x) + (1 ÷ x) ...
POLYNOMIALS 37 Step 5 : We multiply the divisor by the second term of the quotient and subtract the product from the dividend. T ...
38 MATHEMATICS Let us consider some more examples. Example 7 : Divide the polynomial 3x^4 – 4x^3 – 3x –1 by x – 1. Solution : By ...
POLYNOMIALS 39 So, we find that the remainder is 0. Here p(x) = x^3 + 1, and the root of x + 1 = 0 is x = –1. We see that p(–1) ...
40 MATHEMATICS Solution : As you know, q(t) will be a multiple of 2t + 1 only, if 2t + 1 divides q(t) leaving remainder zero. No ...
POLYNOMIALS 41 = –8 + 12 – 10 + 6 =0 So, by the Factor Theorem, x + 2 is a factor of x^3 + 3x^2 + 5x + 6. Again, s(–2) = 2(–2) + ...
42 MATHEMATICS So, let us look for the pairs of factors of 30. Some are 1 and 30, 2 and 15, 3 and 10, 5 and 6. Of these pairs, 2 ...
POLYNOMIALS 43 So, y – 2 is a factor of p(y). Also, p(3) = 3^2 – (5 × 3) + 6 = 0 So, y – 3 is also a factor of y^2 – 5y + 6. The ...
44 MATHEMATICS (i) p(x) = 2x^3 + x^2 – 2x – 1, g(x) = x + 1 (ii) p(x) = x^3 + 3x^2 + 3x + 1, g(x) = x + 2 (iii) p(x) = x^3 – 4x^ ...
POLYNOMIALS 45 Example 17 : Evaluate 105 × 106 without multiplying directly. Solution : 105 × 106 = (100 + 5) × (100 + 6) = (100 ...
46 MATHEMATICS =x^2 + 2xy + y^2 + 2xz + 2yz + z^2 (Using Identity I) =x^2 + y^2 + z^2 + 2xy + 2yz + 2zx (Rearranging the terms) ...
POLYNOMIALS 47 So, we get the following identity: Identity VI : (x + y)^3 =x^3 + y^3 + 3xy (x + y) Also, by replacing y by –y in ...
48 MATHEMATICS Example 24 : Factorise 8x^3 + 27y^3 + 36x^2 y + 54xy^2 Solution : The given expression can be written as (2x)^3 + ...
POLYNOMIALS 49 Expand each of the following, using suitable identities: (i) (x + 2y + 4z)^2 (ii) (2x – y + z)^2 (iii) (–2x + 3y ...
50 MATHEMATICS 16.What are the possible expressions for the dimensions of the cuboids whose volumes are given below? Volume : 3x ...
CHAPTER 3 COORDINATE GEOMETRY What’s the good of Mercator’s North Poles and Equators, Tropics, Zones and Meridian Lines?’ So the ...
52 MATHEMATICS II. Suppose you put a dot on a sheet of paper [Fig.3.2 (a)]. If we ask you to tell us the position of the dot on ...
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