Higher Maths

Polynomials & Quadratics

Use this checklist with the Clelland Maths videos to master synthetic division, discriminants, and graphs for SQA assessments.

1

Polynomial Skills

Synthetic Division
Set up the table correctly (including 0 for missing terms) and calculate the remainder
The Factor Theorem
Show that \( f(h) = 0 \implies (x-h) \) is a factor (Formal statement required)
Finding Unknown Coefficients
Use simultaneous equations to find \( a \) and \( b \) given two pieces of info (e.g. factor and remainder)
Solving Polynomial Equations
Factorise fully then solve \( f(x) = 0 \) to find roots (e.g. \( x = -2, 1, 3 \))
2

Quadratic Theory

Completing the Square
Express functions in the form \( a(x+b)^2 + c \) (including non-unitary coeff of \( x^2 \))
The Discriminant Formula
State and apply \( b^2 - 4ac \) to determine the nature of roots
Nature of Roots Conditions
Know conditions for: Two distinct roots (\(>0\)), Equal roots (\(=0\)), No real roots (\(<0\))
Finding Unknowns with Discriminant
Find range of values for \( k \) or \( p \) that satisfy a specific root condition
3

Graphs & Applications

Quadratic Inequalities
Solve \( ax^2+bx+c > 0 \) by sketching the parabola to find valid intervals
Intersection of Curves
Equate \( y = y \), set to 0, then solve for \( x \) to find points of intersection
Proving Tangency
Show that the discriminant \( b^2 - 4ac = 0 \) to prove a line is a tangent
4

Common Exam Questions

"Show that..." (Factor)
Must write: "Since remainder is 0, \( (x-h) \) is a factor."
"Hence..." (Solve/Factorise)
Use the quadratic quotient from your synthetic division table; do not restart.
Equation from Graph
Use roots to form \( y = k(x-a)(x-b)... \), then sub in a point to find \( k \).
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SQA Exam Tips

  • Communication: You lose the mark if you don't explicitly state the condition for the discriminant (e.g., "For equal roots, \( b^2-4ac=0 \)").
  • Strictly Increasing: If asked to show a function is strictly increasing, show the derivative is a quadratic with \( b^2-4ac < 0 \) and positive \( x^2 \) coeff.
  • Inequalities: Never just guess the range for \( x \). Always sketch the mini-quadratic to see if you need the "inside" or "outside" region.