ELEMENTARY LINEAR ALGEBRA MAP 1099
List 1
Mathematical induction, Newton’s binomial formula,
Complex numbers, Polynomials. 1
8 October 2015
1. Apply mathematical induction to show that following formulas are valid for all n ∈ N:
i) 1 + 3 + · · · + (2n − 1) = n2 ,
ii)
1
1·2
+
1
2·3
+ ··· +
1
n(n+1)
iii) 1 + 3 + · · · + 3n−1 =
iv) 13 + 23 + · · · + n3 =
=
n
,
n+1
3n −1
,
2
h
i
n(n+1) 2
.
2
2. Apply mathematical induction to show that following inequalities are valid for indicated
natural numbers:
i) 2n > n2 for n ­ 5,
ii)
1
12
+
1
22
+ ··· +
1
n2
¬2−
1
n
for n ∈ N,
iii) n! > 2n for n ­ 4,
iv) (1 + x)n ­ 1 + nx for x ­ −1 and n ∈ N (Bernoulli inequality),
v) n! <
n
n
2
for n ­ 6.
3. Apply mathematical induction to show that
i) 5 divides n5 − n for all natural n ­ 2,
ii) 7 divides 8n + 6 for any n ∈ N.
4. Apply Newton’s binomial formula to expand following powers
√
√
√
i) (2x − y)4 , ii) (c + 2)6 , iii) (x + x13 )5 , iv) ( u − 4 v)8 .
15
5. i) Find the coefficient standing in front of a5 in the expansion of a3 + a12 .
√
7
√
4
ii) Find the coefficient standing in front of 4 x in the expansion of
x5 − x33 .
*
6. Comparing real and imaginary part of both sides of given equations find their solutions:
a) z = (2 − i)z,
b) (1 + 3i)z + (2 − 5i)z = 2i − 3.
7. Draw on the complex plane the subsets of C which satisfy following conditions:
i) Re(z + 1) = Im(2z − 4i),
1
ii) Re(z 2 ) = 0,
iii) Im(z 2 ) ¬ 8,
iv) Re( z1 ) > Im(iz).
Translated and compiled from the list of exercises ALGEBRA Z GEOMETRIĄ ANALITYCZNĄ,
ALGEBRA LINIOWA 1 (rok akad. 2015/16) by Jan Goncerzewicz. The original in Polish available on
http://prac.im.pwr.wroc.pl/ gewert/index1.html.
8. Applying geometrical interpretation of the modulus of the difference of two complex
numbers determine and draw the subsets of C which satisfy foloowing conditions:
i) |z + 2 − 3i| < 4,
ii) |z + 5i| ­ |3 − 4i|,
iv) |z + 3i| < |z − 1 − 4i|,
vi)
z−3i z > 1,
vii)
iii) |z − 1| = |1 + 5i − z|,
v) |z + 2 − 3i| < 5,
2 z +4 z−2i vi) |iz + 5 − 2i| < |1 + i|,
viii) |z 2 + 2iz − 1| < 9.
¬ 5,
9. Applying de Moivre’s formula calculate:
√ 6
√
√
√
i) − 21 + i 23 , ii) (2i − 12)9 , iii) ( 5 2 − i 5 2)15 ,
√
iv) ( 3 − i)20 .
Give the answer in algebraic form.
10. Calculate and draw on the complex plane all complex roots of given numbers:
√
√
√
i) 4 −16, ii) 3 27i, iii) 6 8.
11. Solve the following equations in C:
i) z 2 + 4 = 0,
ii) z 2 − 2z + 10 = 0,
v) z 2 + (1 − 3i)z − 2 − i = 0,
iii) z 2 + 3iz + 4 = 0,
vi) z 6 = (1 − i)12 ,
iv) z 4 + 5z 2 + 4 = 0,
vii) (z − i)4 = (z + i)4 .
*
12. Find all integer roots of the following polynomials:
i) x3 + 3x2 − 4,
ii) x4 − 2x3 + x2 − 8x − 12,
iii) x4 − x2 − 2.
13. Find all rational roots of the following polynomials:
i) 12x3 + 8x2 − 3x − 2,
ii) 18x3 − 9x2 − 2x + 1,
iii) 6x4 + 7x2 + 2.
14. Find all roots of given polynomials knowing one of their roots:
i) W (x) = x4 − 4x3 + 12x2 − 16x + 15,
ii) W (x) + x4 − 2x3 + 7x2 + 6x − 30,
√
√
iii) W (x) = x3 − 3 2x2 + 7x − 3 2,
x1 = 1 + 2i,
x1 = 1 − 3i,
√
x1 = 2 + i.
15. For given polynomials P and Q find the reminder in division P by Q without performing
the dividing algorithm (polynomial long division):
i) P (x) = x8 + 3x5 + x2 + 4,
ii) P (x) = x47 + 2x5 − 13,
Q(x) = x2 − 1,
Q(x) = x3 − x2 + x − 1,
iii) P (x) = x99 − 2x98 + 4x97 ,
Q(x) = x4 − 16.
16. Factorize given polynomials into irreducible real polynomials:
i) x3 − 27,
ii) x4 + 16,
iii) x4 + x2 + 1,
iv) x6 − 1.
17. Expand given rational functions into partial fractions:
i)
2x+5
,
x2 −x−2
ii)
x+9
,
x(x+3)2
iii)
3x2 +4x+3
,
x3 −x2 +4x−4
iv)
x3 −2x−7x+6
.
x4 +10x2 +9