Extension Functions – Polynomials

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

• Polynomial terminology
• Reminder and factor theorem
• Sums and product of roots

Content:

MA-F2.1

• Define a general polynomial in one variable, \( x \), of degree \( n \) with real coefficients to be the expression: \( a_nx^n+a_{n−1}x^{n−1}+ \dots +a_2x^2+a_1x+a_0 \), where \( a_n \ne 0 \)
  • understand and use terminology relating to polynomials including degree, leading term, leading coefficient and constant term
• Use division of polynomials to express \( P(x) \) in the form \( P(x)=A(x) \times Q(x)+R(x) \) where \( deg R(x) < deg A(x) \) and \( A(x) \) is a linear or quadratic divisor, \( Q(x) \) the quotient and \( R(x) \) the remainder
  • review the process of division with remainders for integers
  • describe the process of division using the terms: dividend, divisor, quotient, remainder
• Prove and apply the factor theorem and the remainder theorem for polynomials and hence solve simple polynomial equations

MA-F2.2

• Solve problems using the relationships between the roots and coefficients of quadratic, cubic and quartic equations
  • consider quadratic, cubic and quartic equations, and derive formulae as appropriate for the sums and products of roots in terms of the coefficients
• Determine the multiplicity of a root of a polynomial equation
  • prove that if a polynomial equation of the form \( P(x)=0 \) has a root of multiplicity \( r>1 \), then \( P^{\prime}(x)=0 \) has a root of multiplicity \( r−1 \)
• Graph a variety of polynomials and investigate the link between the root of a polynomial equation and the zero on the graph of the related polynomial function
  • examine the sign change of the function and shape of the graph either side of roots of varying multiplicity