C05 Polynomials

对应大纲的内容

MM1.6

Algebraic manipulation of polynomials, including:

expanding brackets and collecting like terms
factorisation and simple algebraic division (by a linear polynomial, including those of the form $ax + b$, and by quadratics, including those of the form $ax^2 + bx + c$ )
use of the Factor Theorem and the Remainder Theorem

MM8.6

Use algebraic techniques to determine where the graph of a function intersects the coordinate axes; appreciate the possible numbers of real roots that a general polynomial can possess.

学习目标

Expand, collect, and factorise complex polynomial expressions.
Perform algebraic division of polynomials by linear divisors (including those of the form $ax+b$ ) and quadratic divisors (including those of the form $a x^2+b x+c$ ).
Understand and expertly apply the Factor Theorem to deduce linear and quadratic factors, and to identify the roots of higher-degree polynomials.
Understand and apply the Remainder Theorem to evaluate the remainders of algebraic division and deduce unknown polynomial coefficients.
Determine the exact intersections between polynomial graphs and the coordinate axes using purely algebraic techniques.
Analyse the real roots of general polynomials, specifically appreciating the maximum possible number of real roots a polynomial of a given degree can possess.