Polynomials - intermediate

This lesson comprises two (2) master classes focusing on:

• Polynomial expressions
• Adding, subtracting, multiplying and dividing polynomials
• Graphing polynomials

Content:

MA5-POL-P-01

Define and operate with polynomials

• Recognise a polynomial expression \( a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 +a_1x + a_0 \) where \( n=0,1,2 \dots \) and \(a_o,a_1,a_2, \dots , a_n \) are real numbers
• Describe polynomials using terms such as degree, leading term, coefficient and leading coefficient, constant term, monic and non-monic
• Define a monic polynomial as having a leading coefficient of one
• Apply the notation \( P(x) \) for polynomials and \( P(c) \) to indicate the value of \( P(x) \) for \( x=c \)
• Add, subtract and multiply polynomials

Divide polynomials

• Identify the dividend, divisor, quotient and remainder in numerical division
• Divide a polynomial by a linear polynomial to find the quotient and remainder
• Express a polynomial in the form \( P(x)=D(x)Q(x)+R(x) \), where \( D(x) \) is the divisor, \( Q(x) \) is the quotient and \( R(x) \) is the remainder

Apply the factor and remainder theorems to solve problems

• Verify the remainder theorem and use it to find factors of polynomials and solve related problems
• Develop and apply the factor theorem to factorise particular polynomials completely and solve related problems
• Apply the factor theorem and division to find the zeroes of a polynomial \( P(x) \) and solve \( P(x)=0\ (degree \le 4) \)
• State the maximum number of zeroes a polynomial of degree \( n \) can have

Graph polynomials

• Graph polynomials in factored form
• Graph quadratic, cubic and quartic polynomials by factorising and finding the zeroes
• Relate the term zeroes to polynomial functions and roots to polynomial equations
• Use graphing applications to determine the effect of single, double and triple roots of a polynomial equation \( P(x)=0 \) on the shape of the graph for \( y=P(x) \)
• Graph polynomials using the sign of the leading term and the multiplicity of roots for the equation \( P(x)=0 \)
• Use graphing applications to compare the graphs of \( y=-P(x) \), \( y=P(-x) \), \( y=P(x)+c \) and \( y=kP(x) \) to the graph of \( y=P(x) \)