Condition for Zeroes of a Cubic Polynomial to Be in Arithmetic Progression Video Explanation Question If the zeroes of the polynomial \[ f(x)=ax^3+3bx^2+3cx+d \] are in arithmetic progression, prove that \[ 2b^3-3abc+a^2d=0. \] Solution Step 1: Assume the Zeroes in A.P. Let the zeroes of the polynomial be \[ \alpha = p-q,\quad \beta = p,\quad […]
Condition for Zeroes of a Cubic Polynomial to Be in Arithmetic Progression Video Explanation Question Find the condition that the zeroes of the polynomial \[ f(x) = x^3 + 3px^2 + 3qx + r \] may be in arithmetic progression. Solution Step 1: Assume the Zeroes in A.P. Let the three zeroes of the polynomial
Finding Zeroes of a Cubic Polynomial in Arithmetic Progression Video Explanation Question If the zeroes of the polynomial \[ f(x) = 2x^3 – 15x^2 + 37x – 30 \] are in arithmetic progression, find them. Solution Step 1: Assume the Zeroes in A.P. Let the three zeroes be \[ a-d,\; a,\; a+d \] where \(a\)
If the Zeroes of the Polynomial f(x) = 2x³ − 15x² + 37x − 30 Are in A.P., Find Them Read More »
Finding the Other Zeroes of a Cubic Polynomial Video Explanation Question If \(4\) is a zero of the cubic polynomial \[ f(x) = x^3 – 3x^2 – 10x + 24, \] find its other two zeroes. Solution Step 1: Verify the Given Zero Substitute \(x = 4\) in \(f(x)\): \[ f(4) = (4)^3 – 3(4)^2
If 4 Is a Zero of the Cubic Polynomial x³ − 3x² − 10x + 24, Find Its Other Two Zeroes Read More »
Finding All Zeroes of a Cubic Polynomial Video Explanation Question Find all the zeroes of the polynomial \[ f(x) = 3x^3 + 10x^2 – 9x – 4, \] if one of its zeroes is \(1\). Solution Step 1: Verify the Given Zero Substitute \(x = 1\) in \(f(x)\): \[ f(1) = 3(1)^3 + 10(1)^2 –
Find All Zeroes of the Polynomial 3x³ + 10x² − 9x − 4, If One of Its Zeroes Is 1 Read More »
Finding a Cubic Polynomial Using Zeros and Their Relationships Video Explanation Question Find the cubic polynomial whose sum of its zeroes is \(3\), sum of the products of its zeroes taken two at a time is \(-1\), product of its zeroes is \(-3\). Solution Step 1: Write the Standard Form of a Cubic Polynomial Let
Verification of Zeros of a Cubic Polynomial Video Explanation Question Verify that the numbers \[ 2,\; 1,\; 1 \] are the zeroes of the cubic polynomial \[ g(x) = x^3 – 4x^2 + 5x – 2, \] and verify the relationship between the zeroes and coefficients. Solution Step 1: Verification of Zeros A number \(x=\alpha\)
Verification of Zeros of a Cubic Polynomial Video Explanation Question Verify that the numbers \[ \frac{1}{2},\; 1,\; -2 \] are the zeroes of the cubic polynomial \[ f(x) = 2x^3 + x^2 – 5x + 2, \] and verify the relationship between the zeroes and coefficients. Solution Step 1: Verification of Zeros A number \(x
Evaluation of an Expression Using Zeros of a Quadratic Polynomial Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the quadratic polynomial \[ f(x) = ax^2 + bx + c, \] evaluate \[ a\!\left(\frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha}\right) + b\!\left(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}\right). \] Solution Step 1: Relations Between Zeros and Coefficients For \( ax^2 +
Evaluation of an Expression Using Zeros of a Quadratic Polynomial Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the quadratic polynomial \[ f(x) = ax^2 + bx + c, \] evaluate \[ \frac{\beta}{a\alpha + b} + \frac{\alpha}{a\beta + b}. \] Solution Step 1: Write Relations Between Zeros and