MAT1511 ASSIGNMENT 2 SEMESTER 1 2025

Description

MAT1511 Assignment 2 FULL ANSWERS Semester 1 | Due April 2025. All questions answered.

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) = x4 + x3 + x2 + x + 12

(Summarize your answer in the form of a table like the example on p. 297). (4)

1.2 P (x) = x4 − 2×3 − 2×2 − 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)

QUESTION 2

Given P (x) = 2×3 − 2×3 − 5×2 − x + 8

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)

QUESTION 3
3.1 Write
(1 + 2i ) (3 + i )
−2 + i
in the form a + bi, where a, b 2 R. (4)
All questions answered.