Higher Maths: polynomials, quadratic theory and recurrence relations

This set covers the algebra side of Higher that sits outside calculus and coordinate geometry: what the factor and remainder theorems let you do with a cubic, how the discriminant is used when a coefficient is unknown, how a quadratic inequality is actually solved, and how a recurrence relation behaves over the long run.

Ten single-answer questions cover the factor theorem, the remainder theorem, factorising a cubic fully, finding the values of k that give equal roots, solving a quadratic inequality, iterating a recurrence relation two steps, finding its limit, stating the condition under which a limit exists at all, completing the square, and reading what a repeated root does to a graph at the x-axis.

Three of the wrong options are worth knowing in advance. A quadratic inequality with an upward parabola is satisfied outside the roots, not between them, and the between-the-roots answer is offered. Equal roots come from a squared term, so both signs of k work and the positive-only answer is offered. And the limit of a recurrence is found by dividing by one minus the multiplier, not by the multiplier itself — that answer is offered too.

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• Use the factor theorem to turn a known root into a linear factor with the right sign
• Use the remainder theorem to find a remainder by evaluation rather than by long division
• Factorise a cubic fully by finding one factor and factorising the quadratic that remains
• Use the discriminant to find the values of an unknown coefficient that give equal roots
• Solve a quadratic inequality by finding the roots and considering which way the parabola opens
• Iterate a recurrence relation the right number of steps
• Find the limit of a recurrence relation by solving the fixed-point equation
• State the condition on the multiplier for a limit to exist
• Complete the square and adjust the constant correctly
• Tell from a factorised polynomial where its graph touches the x-axis and where it cuts

The sequence $u_{n+1} = 0.8u_n + 12$ has a limit $L$. — At the limit, $L = 0.8L + 12$. Rearranging gives $0.2L = 12$, so $L = 12 \div 0.2 = 60$.