Page 1 20 MATHEMA TICS 2 2.1 Introduction In Class IX, you have studied polynomials in one variable and their degrees. 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 3 48 2 uu u ++  is a polynomial in the variable u of degree 6. Expressions like 1 1 x  , 2 x + , 2 1 23 xx ++ etc., are not polynomials. A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 35, x + 2 y + , 2 11 x  , 3z + 4, 2 1 3 u + , 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 , 23 5 xx + y 2 – 2, 2 23, x x + 22 2 21 25,5 ,4 337 u uv vz +  + 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 POL YNOMIALS Page 2 20 MATHEMA TICS 2 2.1 Introduction In Class IX, you have studied polynomials in one variable and their degrees. 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 3 48 2 uu u ++  is a polynomial in the variable u of degree 6. Expressions like 1 1 x  , 2 x + , 2 1 23 xx ++ etc., are not polynomials. A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 35, x + 2 y + , 2 11 x  , 3z + 4, 2 1 3 u + , 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 , 23 5 xx + y 2 – 2, 2 23, x x + 22 2 21 25,5 ,4 337 u uv vz +  + 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 POL YNOMIALS POLYNOMIALS 21 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 replacing 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 that p(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., b k a  =· So, the zero of the linear polynomial ax + b is (Constant term) Coefficient of b ax  = . Thus, the zero of a linear polynomial is related to its coefficients. Does this happen in the case of other polynomials too? For example, are the zeroes of a quadratic polynomial also related to its coefficients? In this chapter, we will try to answer these questions. We will also study the division algorithm for polynomials. 2.2 Geometrical Meaning of the Zeroes of a Polynomial You know that a real number k is a zero of the polynomial p(x) if p(k) = 0. But why are the zeroes of a polynomial so important? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes. Page 3 20 MATHEMA TICS 2 2.1 Introduction In Class IX, you have studied polynomials in one variable and their degrees. 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 3 48 2 uu u ++  is a polynomial in the variable u of degree 6. Expressions like 1 1 x  , 2 x + , 2 1 23 xx ++ etc., are not polynomials. A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 35, x + 2 y + , 2 11 x  , 3z + 4, 2 1 3 u + , 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 , 23 5 xx + y 2 – 2, 2 23, x x + 22 2 21 25,5 ,4 337 u uv vz +  + 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 POL YNOMIALS POLYNOMIALS 21 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 replacing 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 that p(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., b k a  =· So, the zero of the linear polynomial ax + b is (Constant term) Coefficient of b ax  = . Thus, the zero of a linear polynomial is related to its coefficients. Does this happen in the case of other polynomials too? For example, are the zeroes of a quadratic polynomial also related to its coefficients? In this chapter, we will try to answer these questions. We will also study the division algorithm for polynomials. 2.2 Geometrical Meaning of the Zeroes of a Polynomial You know that a real number k is a zero of the polynomial p(x) if p(k) = 0. But why are the zeroes of a polynomial so important? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes. 22 MATHEMA TICS Consider first a linear polynomial ax + b, a ? 0. Y ou have studied in Class IX that the graph of y = ax + b is a straight line. For example, the graph of y = 2x + 3 is a straight line passing through the points (– 2, –1) and (2, 7). x –2 2 y = 2x + 3 –1 7 From Fig. 2.1, you can see that the graph of y = 2x + 3 intersects the xaxis midway between x = –1 and x = – 2, that is, at the point 3 , 0 2 ??  ?? ?? . You also know that the zero of 2x + 3 is 3 2  . Thus, the zero of the polynomial 2x + 3 is the xcoordinate of the point where the graph of y = 2x + 3 intersects the xaxis. In general, for a linear polynomial ax + b, a ? 0, the graph of y = ax + b is a straight line which intersects the xaxis at exactly one point, namely, , 0 b a ?? ?? ?? . Therefore, the linear polynomial ax + b, a ? 0, has exactly one zero, namely, the xcoordinate of the point where the graph of y = ax + b intersects the xaxis. Now, let us look for the geometrical meaning of a zero of a quadratic polynomial. Consider the quadratic polynomial x 2 – 3x – 4. Let us see what the graph* of y = x 2 – 3x – 4 looks like. Let us list a few values of y = x 2 – 3x – 4 corresponding to a few values for x as given in Table 2.1. * Plotting of graphs of quadratic or cubic polynomials is not meant to be done by the students, nor is to be evaluated. Fig. 2.1 Page 4 20 MATHEMA TICS 2 2.1 Introduction In Class IX, you have studied polynomials in one variable and their degrees. 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 3 48 2 uu u ++  is a polynomial in the variable u of degree 6. Expressions like 1 1 x  , 2 x + , 2 1 23 xx ++ etc., are not polynomials. A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 35, x + 2 y + , 2 11 x  , 3z + 4, 2 1 3 u + , 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 , 23 5 xx + y 2 – 2, 2 23, x x + 22 2 21 25,5 ,4 337 u uv vz +  + 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 POL YNOMIALS POLYNOMIALS 21 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 replacing 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 that p(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., b k a  =· So, the zero of the linear polynomial ax + b is (Constant term) Coefficient of b ax  = . Thus, the zero of a linear polynomial is related to its coefficients. Does this happen in the case of other polynomials too? For example, are the zeroes of a quadratic polynomial also related to its coefficients? In this chapter, we will try to answer these questions. We will also study the division algorithm for polynomials. 2.2 Geometrical Meaning of the Zeroes of a Polynomial You know that a real number k is a zero of the polynomial p(x) if p(k) = 0. But why are the zeroes of a polynomial so important? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes. 22 MATHEMA TICS Consider first a linear polynomial ax + b, a ? 0. Y ou have studied in Class IX that the graph of y = ax + b is a straight line. For example, the graph of y = 2x + 3 is a straight line passing through the points (– 2, –1) and (2, 7). x –2 2 y = 2x + 3 –1 7 From Fig. 2.1, you can see that the graph of y = 2x + 3 intersects the xaxis midway between x = –1 and x = – 2, that is, at the point 3 , 0 2 ??  ?? ?? . You also know that the zero of 2x + 3 is 3 2  . Thus, the zero of the polynomial 2x + 3 is the xcoordinate of the point where the graph of y = 2x + 3 intersects the xaxis. In general, for a linear polynomial ax + b, a ? 0, the graph of y = ax + b is a straight line which intersects the xaxis at exactly one point, namely, , 0 b a ?? ?? ?? . Therefore, the linear polynomial ax + b, a ? 0, has exactly one zero, namely, the xcoordinate of the point where the graph of y = ax + b intersects the xaxis. Now, let us look for the geometrical meaning of a zero of a quadratic polynomial. Consider the quadratic polynomial x 2 – 3x – 4. Let us see what the graph* of y = x 2 – 3x – 4 looks like. Let us list a few values of y = x 2 – 3x – 4 corresponding to a few values for x as given in Table 2.1. * Plotting of graphs of quadratic or cubic polynomials is not meant to be done by the students, nor is to be evaluated. Fig. 2.1 POLYNOMIALS 23 Table 2.1 x – 2 –1 012 3 4 5 y = x 2 – 3x – 4 6 0 – 4– 6– 6 – 4 0 6 If we locate the points listed above on a graph paper and draw the graph, it will actually look like the one given in Fig. 2.2. In fact, for any quadratic polynomial ax 2 + bx + c, a ? 0, the graph of the corresponding equation y = ax 2 + bx + c has one of the two shapes either open upwards like or open downwards like depending on whether a > 0 or a < 0. (These curves are called parabolas.) You can see from Table 2.1 that –1 and 4 are zeroes of the quadratic polynomial. Also note from Fig. 2.2 that –1 and 4 are the xcoordinates of the points where the graph of y = x 2 – 3x – 4 intersects the xaxis. Thus, the zeroes of the quadratic polynomial x 2 – 3x – 4 are xcoordinates of the points where the graph of y = x 2 – 3x – 4 intersects the xaxis. This fact is true for any quadratic polynomial, i.e., the zeroes of a quadratic polynomial ax 2 + bx + c, a ? 0, are precisely the xcoordinates of the points where the parabola representing y = ax 2 + bx + c intersects the xaxis. From our observation earlier about the shape of the graph of y = ax 2 + bx + c, the following three cases can happen: Fig. 2.2 Page 5 20 MATHEMA TICS 2 2.1 Introduction In Class IX, you have studied polynomials in one variable and their degrees. 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 3 48 2 uu u ++  is a polynomial in the variable u of degree 6. Expressions like 1 1 x  , 2 x + , 2 1 23 xx ++ etc., are not polynomials. A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 35, x + 2 y + , 2 11 x  , 3z + 4, 2 1 3 u + , 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 , 23 5 xx + y 2 – 2, 2 23, x x + 22 2 21 25,5 ,4 337 u uv vz +  + 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 POL YNOMIALS POLYNOMIALS 21 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 replacing 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 that p(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., b k a  =· So, the zero of the linear polynomial ax + b is (Constant term) Coefficient of b ax  = . Thus, the zero of a linear polynomial is related to its coefficients. Does this happen in the case of other polynomials too? For example, are the zeroes of a quadratic polynomial also related to its coefficients? In this chapter, we will try to answer these questions. We will also study the division algorithm for polynomials. 2.2 Geometrical Meaning of the Zeroes of a Polynomial You know that a real number k is a zero of the polynomial p(x) if p(k) = 0. But why are the zeroes of a polynomial so important? To answer this, first we will see the geometrical representations of linear and quadratic polynomials and the geometrical meaning of their zeroes. 22 MATHEMA TICS Consider first a linear polynomial ax + b, a ? 0. Y ou have studied in Class IX that the graph of y = ax + b is a straight line. For example, the graph of y = 2x + 3 is a straight line passing through the points (– 2, –1) and (2, 7). x –2 2 y = 2x + 3 –1 7 From Fig. 2.1, you can see that the graph of y = 2x + 3 intersects the xaxis midway between x = –1 and x = – 2, that is, at the point 3 , 0 2 ??  ?? ?? . You also know that the zero of 2x + 3 is 3 2  . Thus, the zero of the polynomial 2x + 3 is the xcoordinate of the point where the graph of y = 2x + 3 intersects the xaxis. In general, for a linear polynomial ax + b, a ? 0, the graph of y = ax + b is a straight line which intersects the xaxis at exactly one point, namely, , 0 b a ?? ?? ?? . Therefore, the linear polynomial ax + b, a ? 0, has exactly one zero, namely, the xcoordinate of the point where the graph of y = ax + b intersects the xaxis. Now, let us look for the geometrical meaning of a zero of a quadratic polynomial. Consider the quadratic polynomial x 2 – 3x – 4. Let us see what the graph* of y = x 2 – 3x – 4 looks like. Let us list a few values of y = x 2 – 3x – 4 corresponding to a few values for x as given in Table 2.1. * Plotting of graphs of quadratic or cubic polynomials is not meant to be done by the students, nor is to be evaluated. Fig. 2.1 POLYNOMIALS 23 Table 2.1 x – 2 –1 012 3 4 5 y = x 2 – 3x – 4 6 0 – 4– 6– 6 – 4 0 6 If we locate the points listed above on a graph paper and draw the graph, it will actually look like the one given in Fig. 2.2. In fact, for any quadratic polynomial ax 2 + bx + c, a ? 0, the graph of the corresponding equation y = ax 2 + bx + c has one of the two shapes either open upwards like or open downwards like depending on whether a > 0 or a < 0. (These curves are called parabolas.) You can see from Table 2.1 that –1 and 4 are zeroes of the quadratic polynomial. Also note from Fig. 2.2 that –1 and 4 are the xcoordinates of the points where the graph of y = x 2 – 3x – 4 intersects the xaxis. Thus, the zeroes of the quadratic polynomial x 2 – 3x – 4 are xcoordinates of the points where the graph of y = x 2 – 3x – 4 intersects the xaxis. This fact is true for any quadratic polynomial, i.e., the zeroes of a quadratic polynomial ax 2 + bx + c, a ? 0, are precisely the xcoordinates of the points where the parabola representing y = ax 2 + bx + c intersects the xaxis. From our observation earlier about the shape of the graph of y = ax 2 + bx + c, the following three cases can happen: Fig. 2.2 24 MATHEMA TICS Case (i) : Here, the graph cuts xaxis at two distinct points A and A'. The xcoordinates of A and A' are the two zeroes of the quadratic polynomial ax 2 + bx + c in this case (see Fig. 2.3). Fig. 2.3 Case (ii) : Here, the graph cuts the xaxis at exactly one point, i.e., at two coincident points. So, the two points A and A' of Case (i) coincide here to become one point A (see Fig. 2.4). Fig. 2.4 The xcoordinate of A is the only zero for the quadratic polynomial ax 2 + bx + c in this case.Read More
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