3.1 Algebra

Understand the meaning of |x|, sketch the graph of y = |ax + b|, and use relations such as |a| = |b| ⇔ a² = b² and |x − a| < b ⇔ a − b < x < a + b when solving equations and inequalities.

Graphs of y = |f(x)| and y = f(|x|) for non-linear functions f are not included. Examples: |3x − 2| = |2x + 7|; 2x + 5 < |x + 1|.

Taught in: Modulus: distance from zero, Modulus inequalities: between, or outside · 12 questions

Divide a polynomial of degree not exceeding 4 by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero).

Taught in: Dividing by a quadratic, Dividing polynomials by (x − k), the long way · 10 questions

Use the factor theorem and the remainder theorem.

For example, to find factors and remainders, solve polynomial equations or evaluate unknown coefficients. Includes factors of the form (ax + b) in which the coefficient of x is not unity, and includes calculating remainders.

Taught in: Remainder and factor theorems: substitute instead of dividing, Finding unknown coefficients with the theorems · 11 questions

Recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than (ax + b)(cx + d)(ex + f), (ax + b)(cx + d)² or (ax + b)(cx² + d).

Cases where the degree of the numerator exceeds that of the denominator are excluded.

Taught in: Partial fractions: undoing a single fraction, Partial fractions with repeated and quadratic factors · 13 questions

Use the expansion of (1 + x)ⁿ, where n is a rational number and |x| < 1.

Finding the general term in an expansion is not included. Adapting the standard series to expand, e.g., (2 − ½x)^(−1) is included, as is determining the set of values of x for which the expansion is valid in such cases.