Condition for Divisibility of Polynomials Video Explanation Question If the polynomial \[ f(x) = ax^3 + bx – c \] is divisible by \[ g(x) = x^2 + bx + c, \] find the value of \(ab\). Solution Step 1: Use the Division Algorithm Since \(f(x)\) is divisible by \(g(x)\), \[ f(x) = g(x)\,q(x) \] […]
Finding the Third Zero of a Cubic Polynomial Video Explanation Question If the product of two zeroes of the polynomial \[ f(x) = 2x^3 + 6x^2 – 4x + 9 \] is \(3\), find its third zero. Solution Step 1: Use the Formula for Product of Zeroes For a cubic polynomial \[ ax^3 + bx^2
Condition for Zeroes of a Cubic Polynomial to be in A.P. Video Explanation Question If the zeroes of the polynomial \[ f(x) = x^3 – 3px^2 + qx – r \] are in arithmetic progression, find the required condition. Solution Step 1: Assume the Zeroes in A.P. Let the three zeroes be \[ p-d,\; p,\;
Finding the Value of a Using Product of Zeroes Video Explanation Question If the product of the zeroes of the polynomial \[ f(x) = ax^3 – 6x^2 + 11x – 6 \] is \(4\), find the value of \(a\). Solution Step 1: Use the Formula for Product of Zeroes For a cubic polynomial \[ ax^3
Interpretation of the Graph of a Quadratic Polynomial Given The graph shown in Fig. 2.18 represents the quadratic polynomial \[ f(x) = ax^2 + bx + c \] The coordinates of the vertex are shown as \[ \left(-\frac{b}{2a},\; -\frac{D}{4a}\right) \] where \(D = b^2 – 4ac\) is the discriminant. Solution Step 1: Sign of \(a\)
Interpretation of the Graph of a Quadratic Polynomial Given The graph shown in Fig. 2.17 represents the quadratic polynomial \[ f(x) = ax^2 + bx + c \] The coordinates of the vertex are shown as \[ \left(-\frac{b}{2a},\; -\frac{D}{4a}\right) \] where \(D = b^2 – 4ac\) is the discriminant. Solution Step 1: Sign of \(a\)
If the diagram in Fig. 2.17 shows the graph of the polynomial f(x) = ax^2 + bx + c, then Read More »
Condition for No Real Zeroes of a Quadratic Polynomial Video Explanation Question If the quadratic polynomial \[ f(x) = ax^2 + bx + c \] has no real zeroes and \[ a + b + c < 0, \] then find the sign of \(a\). Solution Step 1: Use the Property of Quadratic Polynomials If
Finding the Value of c Using Zeroes of a Quadratic Polynomial Video Explanation Question If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \[ f(x) = x^2 – p(x+1) – c, \] such that \[ (\alpha + 1)(\beta + 1) = 0, \] find the value of \(c\). Options: (a)
If (α + 1)(β + 1) = 0 for the Polynomial f(x) = x² − p(x + 1) − c, Find the Value of c Read More »
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