Application of Division Algorithm for Polynomials Video Explanation Question Apply the division algorithm to find the quotient and remainder when \[ f(x) = x^3 – 6x^2 + 11x – 6 \] is divided by \[ g(x) = x^2 + x + 1. \] Solution Step 1: Write the Division Algorithm By the division algorithm for […]

Finding the Value of k Using A.P. Condition on Zeroes Video Explanation Question If the zeroes of the polynomial \[ f(x) = x^3 – 12x^2 + 39x + k \] are in arithmetic progression, find the value of \(k\). Solution Step 1: Assume the Zeroes in A.P. Let the three zeroes be \[ a-d,\; a,\;

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 \(a\)

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

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 –

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