If the polynomials \(az^3 + 4z^2 + 3z - 4\) and \(z^3 - 4z + a\) leave the same remainder when divided by \(z - 3\), find the value of \(a\).
\(-1\)
The polynomial \(p(x) = x^4 - 2x^3 + 3x^2 - ax + 3a - 7\) when divided by \(x + 1\) leaves the remainder \(19\). Find the value of \(a\). Also find the remainder when \(p(x)\) is divided by \(x + 2\).
\(a = 5\)
Remainder when divided by \(x + 2\) is \(62\).
If both \(x - 2\) and \(x - \dfrac{1}{2}\) are factors of \(px^2 + 5x + r\), show that \(p = r\).
Without actual division, prove that \(2x^4 - 5x^3 + 2x^2 - x + 2\) is divisible by \(x^2 - 3x + 2\).
Hint: Factorise \(x^2 - 3x + 2\).
Simplify \((2x - 5y)^3 - (2x + 5y)^3\).
\(-120x^2y - 250y^3\)
Multiply \(x^2 + 4y^2 + z^2 + 2xy + xz - 2yz\) by \((-z + x - 2y)\).
\(x^3 - 8y^3 - z^3 - 6xyz\)
If \(a, b, c\) are all non-zero and \(a + b + c = 0\), prove that \(\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab} = 3\).
If \(a + b + c = 5\) and \(ab + bc + ca = 10\), then prove that \(a^3 + b^3 + c^3 - 3abc = -25\).
Prove that \((a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b)(b + c)(c + a)\).