CLASS X MATHS Polynomials
- 2. INTRODUCTION :- • POLYNOMIALS - In mathematics, a polynomial is an expression of finite length constructed from variables and constants with non- negative, whole-number exponents. EG : p (x) = 2x2 + x + 3 p (y) = 6y3 -3y + 5
- 3. TERMS IN A POLYNOMIAL • Monomials – polynomials with one term e.g.:- 4x3 , 7 (constant term) Binomials - polynomials with two terms e.g.:- 3x6 + 13x5 • Trinomials - polynomials with three terms e.g.:- 7x5 - 9 x2 +1 While writing a polynomial always the term with highest exponent is written first . And the other terms are written in descending order according to their powers. EG :- p(x) = 17x9 + 4x7 - 3x4 + 2x2 – 10 .
- 4. DEGREE OF A POLYNOMIAL Degree of a polynomial is the highest among the exponents of all its terms. This highest degree of the polynomial is also called as the order of the polynomial . Examples The degree of the polynomial − 8x7 + 4x3 − 9x + 5 is 7. The degree of the polynomial 5x6 + x4 − 2x3 + 9 is 6. Linear polynomial – polynomial with degree one e.g.:- 2x - 3 , 3z + 4 etc. Quadratic polynomial - polynomial with degree two e.g.:- y2 – 2 , 4z2 + 1 /7 etc. Cubic polynomial - polynomial with degree three e.g.:- x3 + 4x2 – 5 etc.
- 5. ZEROES OF A POLYNOMIAL • ZERO OF A POLYNOMIAL – It is that value of the variable at which the value of the polynomial becomes zero. The zero of the polynomial is defined as any real value of x, for which the value of the polynomial becomes zero. A real number k is a zero of a polynomial p(x) , if p(k) = 0 . EXAMPLE :- Q. Find the zero of the polynomial p(x)=2x +3 . Ans. On equating the polynomial to zero , p(x)=2x +3 = 0 2x = -3 x = -3/2 For checking , we replace x by -3/2 in polynomial p(x)=2x +3 p(-3/2)= 2 (-3/2)+3 = 0 = -3+3 = 0 0=0 Therefore , the zero of the polynomial p(x)=2x +3 is x = -3/2
- 6. NUMBER OF ZEROES OF A POLYNOMIAL The graph of a linear polynomial intersects the x-axis at a maximum of one point. Therefore, a linear polynomial has a maximum of one zero. The graph of a quadratic polynomial intersects the x-axis at a maximum of two points. Therefore, a quadratic polynomial can have a maximum of two zeroes. The graph of a cubic polynomial intersects the x-axis at maximum of three points. A cubic polynomial has a maximum of three zeroes.
- 7. GEOMETRICAL MEANING OF ZEROES OF A POLYNOMIAL Geometrically zeroes of the polynomial are the points where the graph of the polynomial intersects x-axis . If the graph of a polynomial p(x) intersects the x -axis at (k,0), then k is the zero of the polynomial. The equation ax2 + bx + c can have three cases for the graphs.
- 8. Case I :- Here, the graph cuts x-axis at two distinct points A and A′.
- 9. Case (ii):- Here, the graph cuts the x-axis at exactly one point
- 10. Case (iii):- Here, the graph is either completely above the x-axis or completely below the x-axis.
- 11. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A LINEAR POLYNOMIAL • The general form of linear polynomial is p(x)=ax+ b where a≠0 . If its zero is α, then, EXAMPLE Q .Find the zero of the polynomial p(x) = 3x + 4 Ans. We know, for a linear polynomial , the zero α = -b/a . Here , b = 4 and a = 3. Therefore , -b/a = -4/3. Hence the zero of the given polynomial p(x) = 3x + 4 is -4/3.
- 12. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A QUADRATIC POLYNOMIAL • General form of quadratic polynomial is ax2 + bx +c where a≠0. Let the two zeroes of quadratic polynomial p(x)= ax2 + bx +c be α and β . Then,
- 13. Q .Find the sum and product of the zeroes of the polynomial p(x) = 4x2 + 3x +2. Ans. We know, for a quadratic polynomial , α+β = -b/a . αβ = c/a Here , a = 4 , b = 3 and c = 2 Therefore , -b/a = -3/4 and c/a = 2/4 = 1/2 Hence the sum of zeroes of p(x) = 4x2 + 3x +2 is -3/4 and the product of zeroes of p(x) = 4x2 + 3x +2 is 1/2 EXAMPLE
- 14. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A CUBIC POLYNOMIAL • General form of cubic polynomial is ax3 + bx2 + cx + d where a≠0. Then,
- 15. DIVISION ALGORITHM FOR POLYNOMIALS Let f(x), g(x), q(x) and r(x) be polynomials , then the division algorithm for polynomials states that “If f(x) and g(x) are two polynomials such that degree of f(x) is greater that degree of g(x) where g(x) ≠ 0, then there exists two unique polynomials q(x) and r(x) such that f(x) = g(x) x q(x) + r(x) where r(x) = 0 or degree of r(x) less than degree of g(x)”. = f(x) = g(x) x q(x) + r(x) Dividend = divisor x quotient + remainder
- 16. EXAMPLE Consider the cubic polynomial x 3 - 3x 2 - x + 3. If one of its zeroes is 1, then x - 1 is a factor of x 3 - 3x 2 - x + 3. So, you can divide x 3 - 3x 2 - x + 3 by x – 1 and you would get x 2 - 2x – 3 as the quotient . Next, you could get the factors of x 2 - 2x - 3, by splitting the middle term, as : (x + 1)(x - 3). This would give you: x 3 - 3x 2 - x + 3 = (x - 1)(x 2 - 2x - 3) = (x - 1) (x + 1) (x - 3). Now on equating each of these factors to zero , we will get 1, -1, 3. So, all the three zeroes of the cubic polynomial are now known to you as 1, -1, 3.