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Exercise 1.3

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Practice set 1.3 for Complex Numbers (conjugate & division).

Questions

Question 1
Factorize the following .
(i)
$a^2 + 4b^2$
(ii)
$9a^2 + 16b^2$
(iii)
$3x^2 + 3y^2$
(iv)
$144x^2 + 225y^2$
(v)
$z^2 - 2xz - 1$
(vi)
$z^2 + 6z + 13$
(vii)
$z^2 + 4z + 5$
(viii)
$2z^2 - 22z + 65$
Solution Theory
Question 2
Factorize into linear factors.
(i)
$z^3 + 8$
(ii)
$z^3 + 27$
(iii)
$z^3 - 2z^2 + 16z - 32$
(iv)
$z^4 + 21z^2 - 100$
(v)
$z^4 - 16$
(vi)
$z^4 + 3z^2 - 4$
(vii)
$z^4 + 5z^2 + 6$
(viii)
$z^4 - 32z^2 - 3969$
Solution Theory
Question 3
Find roots and express as a product of linear factors: $z^4 + 7z^2 - 144 = 0$ .
Solution Theory
Question 4
Solve complex quadratic equations by completing the square .
(i)
$2z^2 - 3z + 4 = 0$
(ii)
$z^2 - 6z + 30 = 0$
(iii)
$3z^2 - 18z + 50 = 0$
(iv)
$z^2 + 4z + 13 = 0$
(v)
$2z^2 + 6z + 9 = 0$
(vi)
$3z^2 - 5z + 7 = 0$
Solution Theory
Question 5
Solve the following equations .
(i)
$2z^4 - 32 = 0$
(ii)
$3z^5 - 243z = 0$
(iii)
$5z^5 - 5z = 0$
(iv)
$z^3 - 5z^2 + z - 5 = 0$
(v)
$4z^4 - 25z^2 - 21 = 0$
(vi)
$z^3 + z^2 + z + 1 = 0$
Solution Theory
Question 6
Find a polynomial of degree 3 with given zeros and an extra condition .
Solution Theory
Question 7
Find a polynomial of degree 4 with given zeros and an extra condition .
Solution Theory
Question 8
Find a polynomial of degree 4 with given zeros and an extra condition .
Solution Theory