Polynomials (ME-F2)

7 dot points across 7 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

Once the factor theorem links each zero to a factor, what does that force a polynomial to look like: how many zeroes can it have, when is it pinned down by its graph, and how many times can two curves meet?

Develop the structural consequences of the factor theorem: distinct zeroes give distinct linear factors and a degree-n polynomial has at most n zeroes; a polynomial agreeing with another at n + 1 points is identical, so a degree-n graph is fixed by n + 1 points; the number of intersections of two curves is bounded by degree via F(x) = P(x) - Q(x); and re-express a polynomial in powers of (x - a)

The structural consequences of the factor theorem for HSC Maths Extension 1. Distinct zeroes give distinct factors; a degree-n polynomial has at most n zeroes; agreement at n + 1 points forces two polynomials to be identical, so a graph is fixed by n + 1 points; and the difference F = P - Q bounds and locates intersections, with tangency at a double root.

18 min answer →

When a line meets a curve, what does it mean for the line to be a tangent rather than a secant, and how can the sum and product of the roots of one equation locate the point of contact, the midpoint of a chord, and a common tangent without any calculus?

Apply the factor theorem, multiplicity and the sum and product of roots to the geometry of curves: a line is tangent to a curve exactly when the equation formed by solving them simultaneously has a double root; the x-coordinate of the midpoint of a chord is the average of the roots; and the number and nature of the intersections of two curves are read from the roots of the difference of the two polynomials

Geometry of curves through polynomial roots for HSC Maths Extension 1. A line is tangent exactly when solving line and curve gives a double root; a chord midpoint is the average of the roots; intersections are read from the difference of the polynomials. Tangents, chords and common tangents without calculus.

20 min answer →

How do the leading term and the multiplicity of each zero control the shape of a polynomial graph?

Sketch the graph of a polynomial in factored form using the behaviour of the leading term for large x and the multiplicity of each zero, deciding where the curve crosses, touches or has a horizontal inflection, and locate a zero between integers from a table of values

A first-contact answer to the HSC Maths Extension 1 dot point on sketching polynomials. End behaviour from the leading term, the multiplicity rule (cross, touch, horizontal inflection), a stage-by-stage sketch from factors, and locating a zero between integers from a value table, with worked examples.

15 min answer →

How does long division split one polynomial by another into a quotient and a remainder, and what does the identity P(x) = D(x)Q(x) + R(x) tell us?

Divide one polynomial by another using long division, expressing the result in the form P(x) = D(x)Q(x) + R(x) where the remainder has degree less than the divisor, handling missing terms, and writing the result in the rational form P/D = Q + R/D

A first-contact answer to the HSC Maths Extension 1 dot point on dividing polynomials. The division algorithm P(x) = D(x)Q(x) + R(x) with deg R less than deg D, long division by linear and quadratic divisors, handling missing terms with zero coefficients, the remainder form P/D = Q + R/D, and checking by reconstruction, with worked examples.

16 min answer →

How can the remainder and factor theorems tell us the result of a division, and even fully factor a polynomial, without carrying out the long division?

Use the remainder theorem (the remainder on division by x - a is P(a)) and the factor theorem ((x - a) is a factor if and only if P(a) = 0), test divisors of the constant term to locate integer zeroes, and combine these with division to find unknown coefficients and to fully factor a polynomial

A first-contact answer to the HSC Maths Extension 1 dot point on the remainder and factor theorems. The remainder on division by (x - a) is P(a); (x - a) is a factor exactly when P(a) = 0; an integer zero must divide the constant term. Find remainders without dividing, find unknown coefficients, and fully factor a cubic, with worked examples.

17 min answer →

How do the coefficients of a polynomial already know the sum, the product and every symmetric combination of its roots, so that you can answer questions about the roots without ever finding them?

Relate the coefficients of a polynomial to the elementary symmetric functions of its zeroes (with alternating signs) for quadratics, cubics and quartics, and use these relations to find a missing zero, to evaluate symmetric expressions of the zeroes such as the sum of squares and the sum of reciprocals, to find unknown coefficients from a condition on the zeroes, and to handle zeroes in arithmetic or geometric progression or of a special form

First-contact treatment of sums and products of zeroes for HSC Maths Extension 1. The coefficients are the elementary symmetric functions of the roots with alternating signs, for quadratics, cubics and quartics. Use them to find a missing zero and evaluate symmetric expressions without solving the polynomial.

19 min answer →

What exactly is a polynomial, and how do its degree and coefficients control its behaviour?

Define a polynomial and use the language of degree, leading term, leading coefficient, monic and the zero polynomial, including the degree of sums and products and equating coefficients of identically equal polynomials

A first-contact answer to the HSC Maths Extension 1 dot point on the language of polynomials. Degree, leading term and coefficient, monic and the zero polynomial, the degree of sums and products, equating coefficients of identically equal polynomials, and even and odd polynomials, with worked examples.

13 min answer →