Finding All Zeroes of a Polynomial Video Explanation Question Obtain all the zeroes of the polynomial \[ f(x) = 2x^4 + x^3 – 14x^2 – 19x – 6, \] if two of its zeroes are \(-2\) and \(-1\). Solution Step 1: Use the Given Zero \(x = -2\) Since \(-2\) is a zero, \((x + […]
Checking Factor Using Division Algorithm Video Explanation Question Check whether the first polynomial is a factor of the second polynomial: \[ g(x) = 2x^2 – x + 3, \quad f(x) = 6x^5 – x^4 + 4x^3 – 5x^2 – x – 15 \] Solution Step 1: Apply the Division Algorithm By the division algorithm, \[
Checking Factor Using Division Algorithm Video Explanation Question Check whether the first polynomial is a factor of the second polynomial: \[ g(x) = x^3 – 3x + 1, \quad f(x) = x^5 – 4x^3 + x^2 + 3x + 1 \] Solution Step 1: Apply the Division Algorithm By the division algorithm, \[ f(x) =
Checking Factor Using Division Algorithm Video Explanation Question Check whether the first polynomial is a factor of the second polynomial: \[ g(t) = t^2 – 3, \quad f(t) = 2t^4 + 3t^3 – 2t^2 – 9t – 12 \] Solution Step 1: Apply the Division Algorithm By the division algorithm, \[ f(t) = g(t)\,q(t) +
Application of Division Algorithm for Polynomials Video Explanation Question Apply the division algorithm to find the quotient and remainder when \[ f(x) = 15x^3 – 20x^2 + 13x – 12 \] is divided by \[ g(x) = 2 – 2x + x^2. \] Solution Step 1: Arrange the Divisor in Descending Powers \[ g(x) =
Application of Division Algorithm for Polynomials Video Explanation Question Apply the division algorithm to find the quotient and remainder when \[ f(x) = 4x^3 + 8x^2 + 8x + 7 \] is divided by \[ g(x) = 2x^2 – x + 1. \] Solution Step 1: Division Algorithm By the division algorithm, \[ f(x) =
Application of Division Algorithm for Polynomials Video Explanation Question Apply the division algorithm to find the quotient and remainder when \[ f(x) = 10x^4 + 17x^3 – 62x^2 + 30x – 3 \] is divided by \[ g(x) = 2x^2 + 7x + 1. \] Solution Step 1: Division Algorithm By the division algorithm, \[
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