Evaluation Using Zeros of a Quadratic Polynomial Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \[ f(x) = x^2 – p(x+1) – c, \] find \[ (\alpha + 1)(\beta + 1). \] Solution Step 1: Write the Polynomial in Standard Form \[ f(x) = x^2 – […]
Polynomial Whose Zeroes Are Reciprocals of Given Zeroes Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \[ f(x) = x^2 + px + q, \] find the polynomial whose zeroes are \[ \frac{1}{\alpha} \quad \text{and} \quad \frac{1}{\beta}. \] Solution Step 1: Write Relations Between Zeroes and
Finding the Value of k Using Sum of Zeroes Video Explanation Question If the sum of the zeroes of the polynomial \[ f(x) = 2x^3 – 3kx^2 + 4x – 5 \] is \(6\), then find the value of \(k\). Options: (a) 2 (b) 4 (c) -2 (d) -4 Solution Step 1:
Condition for Reciprocal Zeroes of a Quadratic Polynomial Video Explanation Question If one zero of the polynomial \[ f(x) = (k^2 + 4)x^2 + 13x + 4k \] is the reciprocal of the other, find the value of \(k\). Solution Step 1: Use the Condition for Reciprocal Zeroes If the zeroes of a quadratic polynomial
Evaluation Using Zeros of a Quadratic Polynomial Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \[ p(x) = 4x^2 + 3x + 7, \] find \[ \frac{1}{\alpha} + \frac{1}{\beta}. \] Solution Step 1: Write Relations Between Zeroes and Coefficients For the quadratic polynomial \(4x^2 + 3x
Evaluation Using Zeros of a Quadratic Polynomial Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \[ f(x) = x^2 + x + 1, \] find \[ \frac{1}{\alpha} + \frac{1}{\beta}. \] Solution Step 1: Use Relations Between Zeroes and Coefficients For the quadratic polynomial \(x^2 + x
Finding All Zeroes of a Cubic Polynomial Video Explanation Question Given that \(x-\sqrt{5}\) is a factor of the cubic polynomial \[ f(x) = x^3 – 3\sqrt{5}x^2 + 13x – 3\sqrt{5}, \] find all the zeroes of the polynomial. Solution Step 1: Use the Conjugate Factor Theorem Since the coefficients of the polynomial are real and
Finding Other Zeroes of a Cubic Polynomial Video Explanation Question Given that \( \sqrt{2} \) is a zero of the cubic polynomial \[ f(x) = 6x^3 + \sqrt{2}x^2 – 10x – 4\sqrt{2}, \] find its other two zeroes. Solution Step 1: Use the Conjugate Root Theorem Since the coefficients of the polynomial are real and
Making a Polynomial Exactly Divisible Video Explanation Question What must be subtracted from the polynomial \[ f(x) = x^4 + 2x^3 – 13x^2 – 12x + 21 \] so that the resulting polynomial is exactly divisible by \[ g(x) = x^2 – 4x + 3 \; ? \] Solution Step 1: Assume the Remainder Since
Making a Polynomial Exactly Divisible Video Explanation Question What must be added to the polynomial \[ f(x) = x^4 + 2x^3 – 2x^2 + x – 1 \] so that the resulting polynomial is exactly divisible by \[ g(x) = x^2 + 2x – 3 \; ? \] Solution Step 1: Apply the Division Algorithm