Practice Polynomials and Remainder Theorem Practice Questions with Detailed Solutions – Level 3

A polynomial p(x) leaves remainder 5 when divided by \((x - 1)\) and remainder −4 when divided by \((x + 2).\) What is the remainder when p(x) is divided by \((x - 1)(x + 2)?\)

If \((x - 1)\) and \((x + 2)\) are both factors of \(x^{3} + px^{2} + qx + 6,\) what is the value of pq?

If \(\alpha\) and \(\beta\) are the zeroes of the polynomial \(2x^{2} - 5x + 1,\) what is the value of \(\dfrac{\alpha}{\beta} + \dfrac{\beta}{\alpha}?\)

If \(\alpha,\) \(\beta\) and \(\gamma\) are the zeroes of the polynomial \(2x^{3} - 3x^{2} - 11x + 6,\) what is the value of \(\alpha^{2} + \beta^{2} + \gamma^{2}?\)

p(x) is a polynomial of degree 4 and q(x) is a polynomial of degree 3. What is the degree of the polynomial \([p(x)]^{3} \cdot q(x^{2})?\)

The polynomial \(x^{4} - 3x^{3} + 2x^{2} + kx - 5\) leaves remainder 7 when divided by \((x - 2).\) What remainder does it leave when divided by \((x + 1)?\)

What is the HCF of the polynomials \(x^{4} + x^{2} + 1\) and \(x^{3} - 1?\)

What is the LCM of the polynomials \(x^{2} - 5x + 6,\) \(x^{2} - 4\) and \(x^{2} + x - 6?\)

The HCF of two quadratic polynomials is \((x + 1)\) and their LCM is \(x^{3} + 2x^{2} - 5x - 6.\) If one of the polynomials is \(x^{2} - x - 2,\) what is the other?

Which of the following is the complete factorisation of \(x^{3} - 3x^{2} - 10x + 24?\)