Algebra: แบบฝึกหัด

Find the value of the constant \(k\) for which the polynomial \(p(x) = 2x^3 - x^2 + kx - 6\) has \((x - 2)\) as a factor.

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

The polynomial \( P(x) = 2x^4 + ax^3 + bx^2 - 14x + 4 \) is divisible by \( (x^2 - x - 2) \).
Find the values of the constants \( a \) and \( b \).

Find the set of values of \(x\) satisfying the inequality \(|2x - 5| < 7\).

Express the rational function \(\frac{7x - 4}{(x + 2)(x - 3)}\) in partial fractions.

Determine the set of possible values for the constant \( k \) such that the quadratic equation \( 3x^2 - 6x + k = 0 \) has no real roots.

The polynomial \(p(x) = ax^3 - 4x^2 + bx + 18\) is divisible by \((x + 3)\). When it is divided by \((x - 2)\), the remainder is 10. Find the values of the constants \(a\) and \(b\).

Express the rational function \( \frac{3x^2 - x + 10}{(x - 2)(x^2 + 4)} \) in partial fractions, and hence find the first three terms of its binomial expansion in ascending powers of \( x \), simplifying the coefficients. State the range of values of \( x \) for which the expansion is valid.

The polynomial \( p(x) = 2x^3 + ax^2 + bx + 3 \) has a factor \( (x + 3) \). The remainder when \( p(x) \) is divided by \( (x - 2) \) is \( 25 \).
Find the values of the constants \( a \) and \( b \).

(a) Solve the equation \( |2x - 3| = |x + 4| \).
(b) Hence, or otherwise, determine the set of values of \( x \) for which \( |2x - 3| \geq |x + 4| \).