More about polynomials：練習問題

\(\frac{3}{5k+2} - \frac{2}{5k-2} =\)

Let \( P(x) = x^3 + ax^2 + bx + 5 \), where \( a \) and \( b \) are constants. If \( x-1 \) is a factor of \( P(x) \) and the remainder when \( P(x) \) is divided by \( x+1 \) is \( 12 \), find the value of \( P(2) \).

The H.C.F. and the L.C.M. of three expressions are \(mn^3\) and \(60m^5n^4p^2\) respectively. If the first expression and the second expression are \(10m^4n^3p\) and \(12m^2n^4p^2\) respectively, which of the following could be the third expression?

Simplify the rational function \(\frac{x^2 - 9}{x+3}\), where \(x \neq -3\).

Suppose the polynomial \(P(x) = x^3 + ax^2 + bx - 6\) is divisible by \(x^2 + x - 2\). Find the value of \(a - b\).

Find the remainder when the polynomial \(P(x) = x^3 + 2x^2 - x + 5\) is divided by \(x + 1\).

Suppose that the polynomial \(P(x)\) is divided by \(x^2 + 5x + 6\), the remainder is \(2x - 3\). Find the remainder when \(P(x)\) is divided by \(x + 2\).

Let \(P(x) = x^3 + ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. It is given that the greatest common divisor (GCD) of \(P(x)\) and \(x^2 - 4\) is \(x+2\). When \(P(x)\) is divided by \(x+1\), the remainder is 6. Given \(P(2) = 12\), find the values of \(a\), \(b\), and \(c\).

Let \(P(x) = x^3 + kx^2 - 4x - 4\), where \(k\) is a constant.

(a) If \((x + 2)\) is a factor of \(P(x)\), find the value of \(k\).

(b) Factorize \(P(x)\) completely.

(c) Simplify the rational function \(\frac{x^2 - x - 2}{P(x)}\).

Let \(P(x) = 3x^3 + ax^2 + bx + 12\), where \(a\) and \(b\) are constants. When \(P(x)\) is divided by \(x^2 - x - 2\), the remainder is \(5x + 10\).

(a) Find the values of \(a\) and \(b\).

(b) Let \(Q(x) = P(x) - (5x + 10)\). Factorize \(Q(x)\) completely.

(c) Let \(S(x) = 3x^2 + 5x - 2\). Find the least common multiple (LCM) of \(Q(x)\) and \(S(x)\).

(d) Simplify the rational function \(\frac{S(x)}{Q(x)} + \frac{1}{x^2 - x - 2}\).