Factors of polynomials：練習問題

The polynomial \(f(x) = x^3 + ax^2 + bx - 6\) has a factor \((x - 2)\). When \(f(x)\) is divided by \((x - 1)\), the remainder is \(-6\). Find the values of the constants \(a\) and \(b\).

Given that \((2x+1)\) is a factor of \(P(x) = 4x^3 + ax^2 - 7x - 2\), find the value of \(a\).

A polynomial \(P(x)\) is such that when it is divided by \((x-3)\), the remainder is 7. When \(P(x)\) is divided by \((x+1)\), the remainder is -1. Find the remainder when \(P(x)\) is divided by \((x-3)(x+1)\).

The polynomial \(P(x) = ax^3 + bx^2 - 5x + 2\) leaves a remainder of \(2x + 4\) when divided by \(x^2 - 1\). Determine the values of the constants \(a\) and \(b\).

If \( (x-3) \) is a factor of \( f(x) = x^2 + kx - 15 \), find the value of \( k \).

When the polynomial \(P(x) = x^3 + ax^2 + bx - 6\) is divided by \((x-1)\), the remainder is \(-4\). When divided by \((x+1)\), the remainder is \(-12\). Determine the values of \(a\) and \(b\).

The polynomial \(R(x) = 2x^3 - 3x^2 + px + q\) has a remainder of 4 when divided by \((x-1)\) and a remainder of \(-2\) when divided by \((2x+1)\). Find the values of \(p\) and \(q\).

Given the polynomial \( P(x) = 2x^3 - kx^2 + 5x - 3 \). When \( P(x) \) is divided by \( (x-2) \), the remainder is 7. Find the value of \( k \).

Given that \( g(x) = 2x^3 - x^2 - 13x - 6 \).
(a) Use the Factor Theorem to show that \( (2x+1) \) is a factor of \( g(x) \).
(b) Hence, factorise \( g(x) \) completely.