The factor and remainder theorems, factorising a cubic, equal roots, quadratic inequalities, completing the square, and the limit of a recurrence…
10 questions • Zestly
• Given the polynomial $f(x) = x^3 - 3x^2 + 4$, it is found that $f(2) = 0$. What does this result tell you about the polynomial?
• What is the remainder when $x^3 + 2x^2 - 5x + 1$ is divided by $(x - 3)$?
• Factorise $x^3 - 3x^2 - 4x + 12$ fully.
• For which values of $k$ does the equation $x^2 + kx + 9 = 0$ have equal roots?
• Solve the inequality $x^2 - x - 6 > 0$.
• A sequence is defined by $u_{n+1} = 0.8u_n + 12$ with $u_0 = 50$. What is the value of $u_2$?
• The sequence $u_{n+1} = 0.8u_n + 12$ has a limit $L$ as $n$ tends to infinity. What is the value of this limit?
• Why does a recurrence relation of the form $u_{n+1} = au_n + b$ have a limit as $n$ tends to infinity?
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