HONG KONG DIPLOMA OF SECONDARY EDUCATION
EXAMINATION
MATHEMATICS
Compulsory Part
SCHOOL-BASED ASSESSMENT
Sample Assessment Task
Polynomial
教育局 課程發展處 數學教育組
Mathematics Education Section, Curriculum Development Institute
The Education Bureau of the HKSAR
Instructions
1.
This task consists of FOUR parts, Part A to Part D. Students are required to attempt ALL parts.
2.
Numerical answers should be exact or correct to 3 significant figures.
3.
All working must be clearly shown.
Teacher Guidelines
1.
This task requires students to
(a)
use factor theorem;
(b)
find the possible polynomial with integral coefficients.
2.
It is anticipated that in general students can complete the task in 1.5 hours. However, teachers can
exercise their professional judgment to adjust the time allowed to cater for the needs of their
students.
3.
Teachers may feel free to modify the question and the marking guidelines to cater for the ability
level of their students.
4.
Feedback should be provided to students after marking the task. For instance, teachers can discuss
different approaches of handling each part of the task with students.
Polynomial
2
Part A
1.
(a)
If f (1)  0 , write a polynomial f ( x)  ax  b where a and b are integers.
(b)
If f (2)  0 , write a polynomial f ( x)  ax  b where a and b are integers.
(c)
2
If f ( )  0 , write a polynomial f ( x)  ax  b where a and b are integers.
3
(d)
2
If f (  )  0 , write a polynomial f ( x)  ax  b where a and b are integers.
3
Part B
2.
(a)
(b)
(i)
If f ( 2 )  0 , find a polynomial f ( x)  ax 2  bx  c where a , b and c are integers.
(ii)
Find another factor of f ( x) in (a)(i).
If f ( x)  x 2  2 x  1 ,
(i)
show that f (1  2 )  0 ;
(ii)
solve f ( x)  0 ;
(iii) write down the factors of f (x) .
(c)
Polynomial
If f (
2 3
)  0 and f ( x)  ax 2  bx  c where a , b and c are integers,
3
(i)
write another factor of f ( x)  ax 2  bx  c ;
(ii)
hence, find the polynomial f ( x)  ax 2  bx  c .
3
Part C
3.
After teaching the topic “Polynomial”, teacher asked students to complete a challenging question
at home. Chi Keung finished the question within few minutes. He was very happy and sent the full
solution to his best friend Wai Yin for comments.
Question :
If f (1  2 )  0 and f (1  2 )  0 , find a polynomial f (x) with the least degree
and integral coefficients.
Solution :
 f (1  2 )  0 and f (1  2 )  0 ,

 x  1  2  and  x  1  2  are factors of f (x) .
2
 x  1  2  x  1  2   x2  1  2   x2  1  2  x2  3
Hence, f ( x)  x 2  3 .
If you were Wai Yin, would you accept Chi Keung’s solution? If not, explain and give a correct
solution.
Part D
4.
Chi Keung finds the following question from the past papers of the International Mathematics
Competition.
Find a polynomial f (x) with the least possible degree and integral coefficients such that
f ( 6  3  2)  0 .
He knows that the working is similar to the one for the challenging question just finished in
Question 3 above, but the degree of the polynomial should be 8. Could you answer the question on
his behalf ?
END OF ASSESSMENT TASK
Polynomial
4