One Shot: बहुपद (polynomials)| Class 10 Maths Chapter 2 | Full Chapter Revision by Maneesh Academy скачать в хорошем качестве

One Shot: बहुपद (polynomials)| Class 10 Maths Chapter 2 | Full Chapter Revision by Maneesh Academy

One Shot: बहुपद (polynomials)| Class 10 Maths Chapter 1 | Full Chapter Revision POLYNOMIALS An expression of the form p(x)= a_{0} + a_{1}*x + a_{2} * x ^ 2 +...+a n x^ n , where a_{n} ne0, is called a polynomial in x of degree n., Here a_{0}, a_{1}, a_{2} ,...,a n are real numbers and each power of x is a non-negative integer. LINEAR POLYNOMIAL A polynomial of degree 1 is called a linear polynomial. A linear polynomial is of the form p(x) = ax + b where a ne0 . QUADRATIC POLYNOMIAL A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial is of the form p(x) = a * x ^ 2 + bx + c where a ne0. CUBIC POLYNOMIAL A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial is of the form p(x) = a * x ^ 3 + b * x ^ 2 + cx + d where a ne0. BIQUADRATIC POLYNOMIAL A polynomial of degree 4 is called a biquadratic polynomial. VALUE OF A POLYNOMIAL AT A GIVEN POINT If p(x) is a polynomial in x and if a is any real number then the value obtained by putting x = alpha in p(x) is called the value of p(x) at x = alpha The value of p(x) at x = a is denoted by p(alpha) ZEROS OF A POLYNOMIAL A real number a is called a zero of the polynomial p(x) if p(alpha) = 0 RELATION BETWEEN THE ZEROS AND COEFFICIENTS OF A QUADRATIC POLYNOMIAL Let a and ẞ be the zeros of a quadratic polynomial p(x) = a * x ^ 2 + bx + c where a ne0, Then, (x - alpha) and (x - beta) are the factors of p(x) (a * x ^ 2 + bx + c) = k(x - alpha)(x - beta) where k is a constant = k\{x ^ 2 - (alpha + beta) * x + alpha*beta\} = k * x ^ 2 - k(alpha + beta) * x + k(alpha*beta) Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients: 1. x ^ 2 + 7x + 12 2. x ^ 2 - 2x - 8 3. x ^ 2 + 3x - 10 4. 4x ^ 2 - 4x - 3 5.x ^ 2 - 4 - 8x 6. 2sqrt(3) * x ^ 2 - 5x + sqrt(3) 7. 2x ^ 2 - 11x + 15 8. 4x ^ 2 - 4x + 1 13. Find the quadratic polynomial whose zeros are 2 and 6. Verify the relation between the coefficients and the zeros of the polynomial. 14. Find the quadratic polynomial whose zeros are 2/3 and - 1/4 Verify the relation between the coefficients and the zeros of the polynomial. 15. Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial. 16. Find the quadratic polynomial, the sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial. 17. Find the quadratic polynomial, the sum of whose zeros is (5/2) and their product is 1. Hence, find the zeros of the polynomial. 18. Find the quadratic polynomial, the sum of whose roots is sqrt(2) and their product is HB-52 1/3 19. If the quadratic equation a * x ^ 2 + 7x + b = 0 and x = - 3 are the roots of the then find the values of a and b. x = 2/3 20. If (x + a) is a factor of the polynomial 2x ^ 2 + 2ax + 5x + 10 find the value of a. is 2/3 Find the other 21. One zero of the polynomial 3x ^ 3 + 16x ^ 2 + 15x - 18 is zeros of the polynomial. RELATION BETWEEN THE ZEROS AND COEFFICIENTS OF A CUBIC POLYNOMIAL Let a, ẞ and y be the zeros of a cubic polynomial p(x) = a * x ^ 3 + b * x ^ 2 + cx + d where a ne0. DIVISION ALGORITHM FOR POLYNOMIALS If f(x) and g(x) are any two polynomials with g(x) ne0 then we can find polynomials q(x) and r(x) such that f(x) = q(x) * g(x) + r(x) 1. Verify that 3, 2, 1 are the zeros of the cubic polynomial p(x) = x ^ 3 - 2x ^ 2 - 5x + 6 and verify the relation between its zeros and coefficients. 2. Verify that 5,-2 and 1/3 are the zeros of the cubic polynomial p(x) = 3x ^ 3 - 10x ^ 2 - 27x + 10 and verify the relation between its zeros and coefficients. 3. Find a cubic polynomial whose zeros are 2, -3 and 4. 4. Find a cubic polynomial whose zeros are 1/(2') and -3. 5. Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, -2 and -24 respectively. Find the quotient and the remainder when: 6. f(x) = x ^ 3 - 3x ^ 2 + 5x - 3 is divided by g(x) = x ^ 2 - 2 7. f(x) = x ^ 4 - 3x ^ 2 + 4x + 5 is divided by g(x) = x ^ 2 + 1 - x 8. f(x) = x ^ 4 - 5x + 6 is divided by g(x) = 2 - x ^ 2 9. By actual division, show that x ^ 2 - 3 is a factor of 2x ^ 4 + 3x ^ 3 - 2x ^ 2 - 9x - 12 10. On dividing 3x ^ 3 + x ^ 2 + 2x + 5 by a polynomial g(x), the quotient and remainder are (3x - 5) and (9x + 10) respectively. Find g(x). g(x) = ((3x ^ 3 + x ^ 2 + 2x + 5) - (9x + 10))/(3x - 5) 11. Verify division algorithm for the polynomials f(x) = 8 + 20x + x ^ 2 - 6x ^ 3 and g(x) = 2 + 5x - 3x ^ 2 12. It is given that -1 is one of the zeros of the polynomial x ^ 3 + 2x ^ 2 - 11x - 12 Find all the zeros of the given polynomial.