Mathematics IM
Worked Examples
ALGEBRA: POLYNOMIAL FUNCTIONS
Produced by the Maths Learning Centre, The University of Adelaide.
May 3, 2013
The questions on this page have worked solutions and links to videos on the following
pages. Click on the link with each question to go straight to the relevant page. You will
need to have the question handy to refer to while watching the videos.
Questions
1. See Page 2 for worked solutions.
For each of the following quadratic equations find the y-intercept, the axis of symmetry,
the vertex and the zeros and sketch the function, showing each of these features on
your graph. (a) y = 3x2 + 2x − 8
(b) y = 2x2 + 4x + 5
2. See Page 4 for worked solutions.
Find the y-intercept, symmetry point and zeros of the cubic equation
y = x3 − x2 + x − 1 = (x − 1)(x2 + 1). Sketch the graph of this function.
3. See Page 5 for worked solutions.
Consider P (x) = x4 − 3x3 − 2x2 + 12x − 8.
(a) Use the factor theorem to show that (x − 1) and (x + 2) are factors of this polynomial.
(b) Divide P (x) by an appropriate polynomial to completely factorise it and hence
find all of its zeros.
1
1. Click here to go to question list.
For each of the following quadratic equations find the y-intercept, the axis of symmetry,
the vertex and the zeros and sketch the function, showing each of these features on
your graph. (a) y = 3x2 + 2x − 8
(b) y = 2x2 + 4x + 5
Click here to see video of this example on YouTube.
2
3
2. Click here to go to question list.
Find the y-intercept, symmetry point and zeros of the cubic equation
y = x3 − x2 + x − 1 = (x − 1)(x2 + 1). Sketch the graph of this function.
Click here to see video of this example on YouTube.
4
3. Click here to go to question list.
Consider P (x) = x4 − 3x3 − 2x2 + 12x − 8.
(a) Use the factor theorem to show that (x − 1) and (x + 2) are factors of this polynomial.
(b) Divide P (x) by an appropriate polynomial to completely factorise it and hence
find all of its zeros.
Click here to see video of this example on YouTube.
5
6