MAT1511
ASSIGNMENT 2 2025
UNIQUE NO.
DUE DATE: 2025
, lOMoARcPSD|21997160
INSTRUCTIONS TO CANDIDATES:
• The use of a pocket calculator is NOT permissible.
• Answer ALL the questions.
• Show ALL your workings.
QUESTION 1
1.1 Use Descartes’ Rule of Signs to describe all possibilities for the number of positive, negative
and imaginary zeros of
P (x) = x 4 + x 3 + x 2 + x + 12
(Summarize your answer in the form of a table like the example on p. 297). (4)
1.2 P (x) = x 4 − 2x 3 − 2x 2 − 2x − 3
(a) Use the Upper and Lower Bounds Theorem to show that all zero of P (x) are bounded
below by −1 and above by 3. (3)
(b) Find all the possible rational zeros of P(x) by using the Rational Zero Theorem. (1)
(c) Solve P (x) = 0 (i.e. find all the solutions of P(x) = 0.) (3)
[11]
QUESTION 2
Given P (x) = 2x − 2x − 5x − x + 8
3 3 2
2.1 Use the Upper and Lower Bounds Theorem to show that all zero of P (x) are bounded below
by −1 and above by 3. (4)
2.2 Use the Rational Zero Theorem and Factor Theorem to solve P (x) = 0 (i.e. find all the
solutions of P (x) = 0.) (5)
[9]
QUESTION 3
3.1 Write
(1 + 2i) (3 + i)
−2 + i
in the form a + bi, where a, b ∈ R. (4)
3.2 Decompose
4x 2 − 14x + 2
4x 2 − 1
into partial fractions (show all the steps). (6)
[10]
2
ASSIGNMENT 2 2025
UNIQUE NO.
DUE DATE: 2025
, lOMoARcPSD|21997160
INSTRUCTIONS TO CANDIDATES:
• The use of a pocket calculator is NOT permissible.
• Answer ALL the questions.
• Show ALL your workings.
QUESTION 1
1.1 Use Descartes’ Rule of Signs to describe all possibilities for the number of positive, negative
and imaginary zeros of
P (x) = x 4 + x 3 + x 2 + x + 12
(Summarize your answer in the form of a table like the example on p. 297). (4)
1.2 P (x) = x 4 − 2x 3 − 2x 2 − 2x − 3
(a) Use the Upper and Lower Bounds Theorem to show that all zero of P (x) are bounded
below by −1 and above by 3. (3)
(b) Find all the possible rational zeros of P(x) by using the Rational Zero Theorem. (1)
(c) Solve P (x) = 0 (i.e. find all the solutions of P(x) = 0.) (3)
[11]
QUESTION 2
Given P (x) = 2x − 2x − 5x − x + 8
3 3 2
2.1 Use the Upper and Lower Bounds Theorem to show that all zero of P (x) are bounded below
by −1 and above by 3. (4)
2.2 Use the Rational Zero Theorem and Factor Theorem to solve P (x) = 0 (i.e. find all the
solutions of P (x) = 0.) (5)
[9]
QUESTION 3
3.1 Write
(1 + 2i) (3 + i)
−2 + i
in the form a + bi, where a, b ∈ R. (4)
3.2 Decompose
4x 2 − 14x + 2
4x 2 − 1
into partial fractions (show all the steps). (6)
[10]
2
