Class 10th chapter-2 (polynomials) बहुपद part - 6

For Class 10, the most common questions on polynomials from previous year's papers (PYQs) revolve around a few key concepts. Here’s a breakdown of those topics and some example questions. 1. Finding Zeros of a Quadratic Polynomial This is a fundamental skill. You are given a quadratic polynomial and asked to find its zeros (roots) by factorization or using the quadratic formula. Example 1: Find the zeros of the quadratic polynomial 6x^2 - 3 - 7x and verify the relationship between the zeros and the coefficients. Solution Approach: Rewrite the polynomial in standard form: 6x^2 - 7x - 3. Factorize the polynomial. Find two numbers that multiply to 6 \times (-3) = -18 and add up to -7. These are -9 and 2. Split the middle term: 6x^2 - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (3x + 1)(2x - 3). Set the factors to zero to find the roots: 3x + 1 = 0 \Rightarrow x = -1/3 and 2x - 3 = 0 \Rightarrow x = 3/2. Verification: Sum of zeros: \alpha + \beta = (-1/3) + (3/2) = (-2 + 9)/6 = 7/6. From the polynomial, the sum of zeros is -b/a = -(-7)/6 = 7/6. Product of zeros: \alpha\beta = (-1/3) \times (3/2) = -3/6 = -1/2. From the polynomial, the product of zeros is c/a = -3/6 = -1/2. Since both relationships hold true, the verification is complete. 2. Relationship Between Zeros and Coefficients Questions often test your understanding of the formulas relating the zeros of a polynomial to its coefficients. For a quadratic polynomial ax^2 + bx + c, with zeros \alpha and \beta: Sum of zeros: \alpha + \beta = -b/a Product of zeros: \alpha\beta = c/a For a cubic polynomial ax^3 + bx^2 + cx + d, with zeros \alpha, \beta, \gamma: Sum of zeros: \alpha + \beta + \gamma = -b/a Sum of products of zeros taken two at a time: \alpha\beta + \beta\gamma + \gamma\alpha = c/a Product of zeros: \alpha\beta\gamma = -d/a Example 2: If \alpha and \beta are the zeros of the quadratic polynomial f(x) = x^2 - x - 4, find the value of \frac{1}{\alpha} + \frac{1}{\beta}. Solution: From the polynomial, a=1, b=-1, c=-4. Find the sum and product of the zeros: \alpha + \beta = -b/a = -(-1)/1 = 1. \alpha\beta = c/a = -4/1 = -4. Now, simplify the expression: \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta}. Substitute the values: \frac{1}{\alpha} + \frac{1}{\beta} = \frac{1}{-4} = -1/4. 3. Finding a Polynomial Given Its Zeros You are given the zeros or their sum and product, and you have to form the polynomial. The formula for a quadratic polynomial with zeros \alpha and \beta is: p(x) = k[x^2 - (\text{sum of zeros})x + (\text{product of zeros})] p(x) = k[x^2 - (\alpha + \beta)x + \alpha\beta], where k is any non-zero real number. Example 3: Find the quadratic polynomial, the sum of whose zeros is -5 and their product is 6. Solution: Sum of zeros (\alpha + \beta) = -5. Product of zeros (\alpha\beta) = 6. Using the formula: p(x) = x^2 - (\text{sum})x + (\text{product}) = x^2 - (-5)x + 6 = x^2 + 5x + 6. 4. Problems on Division Algorithm for Polynomials This involves using the formula p(x) = g(x) \cdot q(x) + r(x) to solve for an unknown polynomial or a specific value. Example 4: If the polynomial x^4 - 6x^3 + 16x^2 - 25x + 10 is divided by another polynomial x^2 - 2x + k, the remainder comes out to be x + a, find the values of k and a. Solution: This type of question requires you to perform polynomial long division. After performing the division, you will get a remainder. By comparing this remainder with the given remainder x+a, you can solve for the unknown constants k and a. 5. Graphing of Polynomials Questions might ask you to find the number of zeros of a polynomial from its graph. The number of zeros is equal to the number of times the graph intersects the x-axis. Example 5: A graph of a polynomial y = p(x) is given. Find the number of zeros of p(x). Solution: Simply count the number of points where the curve crosses or touches the x-axis. Each point of intersection represents a zero.