Higher Maths

Polynomials & Quadratics

Master synthetic division, the discriminant, completing the square, and solving polynomial equations with this essential guide.

1

Polynomials & Division

Synthetic Division
Use coefficients of the polynomial. If a term is missing (e.g., no \(x^2\)), put a 0 in the table.
The Remainder Theorem
When \(f(x)\) is divided by \((x-h)\), the remainder is \(f(h)\).
The Factor Theorem
If \(f(h) = 0\) (Remainder is 0), then \((x-h)\) is a factor.
Solving Equations
Find one factor first, divide to get a quadratic, then factorise the quadratic fully.
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The Quadratic Discriminant

The Formula
For \(ax^2 + bx + c = 0\), the discriminant is: \( b^2 - 4ac \)
Two Real Distinct Roots
Condition: \( b^2 - 4ac > 0 \)
Equal Roots (One Repeated Root)
Condition: \( b^2 - 4ac = 0 \) (Tangent to axis)
No Real Roots
Condition: \( b^2 - 4ac < 0 \) (Does not cross x-axis)
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Graphs, Shapes & Forms

Completing the Square
Form: \( a(x+p)^2 + q \).
Turning point is at \( (-p, q) \).
Intersection of Curves
Equate the functions: \( f(x) = g(x) \).
Rearrange to \( = 0 \) and solve for \( x \).
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Finding Unknowns
If given 2 pieces of info (e.g., a factor and a remainder), set up simultaneous equations to find \( a \) and \( b \).
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SQA Exam Tips

  • The "Statement": For "Show that..." questions, you must write a conclusion: "Since remainder is 0, (x-k) is a factor."
  • "Hence": If part (b) says "Hence", use the quadratic result from your synthetic division in part (a). Do not start from scratch!
  • Inequalities: When solving \( ax^2+bx+c > 0 \), always sketch the graph to see where the curve is above or below the axis.