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2020 HSC Mathematics Extension 1

Question 5:
Polynomial graphs

1 markpolynomial graphs

A monic polynomial of degree has one repeated zero of multiplicity and is divisible by .

Which of the following could be the graph of ?

The four quartic graph options supplied with the question.
The four quartic graph options supplied with the question.
  1. Downward-opening graph with two distinct real zeros, both crossed
  2. Downward-opening graph with a touch and a separate cross
  3. Upward-opening graph that touches the -axis at exactly one real zero (double root) and has no other real zeros
  4. Upward-opening graph that crosses the -axis at two distinct real zeros

How to recognise this question

Question type: polynomial graphs

  • The command and notation point to polynomial graphs.
  • The requested response is a multiple choice.

How to handle it: use degree, leading coefficient, root multiplicities and any forced factors to decide shape.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. Monic of degree means leading coefficient , so both ends go to (opens upwards). Eliminate A and B.
  2. Divisible by , which has discriminant , so that quadratic factor has no real zeros.
  3. The only real zero is the given double zero. Even multiplicity means the graph touches the -axis and turns back without crossing.
  4. Only option C matches: upward ends and a single touch-point real zero.
  5. Answer: C.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.

Try your hand at it again by answering these questions

Question 1

A monic quartic is divisible by and has a double real zero at . Which statement is true?

  1. It touches the -axis only at and both ends go to
  2. It crosses at and both ends go to
  3. It has four distinct real zeros
  4. It is undefined at
See the step-by-step answer guide →

Question 2

A monic quartic has a double zero at and is divisible by (no real zeros). Which graph feature is forced?

  1. Touches at only; both ends
  2. Crosses at ; both ends
  3. Two simple real zeros only
  4. A vertical asymptote at
See the step-by-step answer guide →

Question 3

Which polynomial matches “monic quartic, double zero at , divisible by ”?

See the step-by-step answer guide →