Answer to Question #98548 in Algebra for Shaheli Chattopadhyay

Question

(a) Solve the following linear systems by the
process of elimination : 3
x + 2y + 5z = 9
x — y + $z = 2
3x — 6y z = 26
(b) Give an example, with justification, of each of
the following : 2
(i) An element of (C x Q) \(Q x C);
(ii) An infinite set whose complement in R
is infinite.
2. (a) Find the roots of the polynomial equation
x4— 9x3+ 17x2+ 33x — 90 = 0,
given that two of its roots are equal to 3. 2
(b) Use the principle of mathematical induction
to show that for any positive integer n,
(311— 2n) > O.
(b) Use Cardano's method to obtain the roots of
x3— 3x + 2 = 0.

Answer 1

(a) Solve the following linear systems by the

process of elimination : 3

(1) x + 2y + 5z = 9

(2) x — y + $z = 2

(3) 3x — 6y z = 26 

(there is a typo, so lets say $z is 4z in equation (2) and 

equation (3) is 3x - 6y + 2z = 26)

(1) x + 2y + 5z = 9

(2) x — y + 4z = 2

(3) 3x — 6y + 2z = 26

1. Choose a variable to eliminate, say x, and select two equations with which to eliminate it, say equations (1) and (2)

x + 2y + 5z = 9

x — y + 4z = 2 (multiply by -1 and add 2 equations)

x + 2y + 5z = 9

-x + y - 4z = -2

____________

3y + z = 7 (4)

2. Select a different set of two equations, say equations (2) and (3), and eliminate the same variable.

x - y + 4z = 2 (multiply by -3 and add 2 equations)

3x - 6y + 2z = 26

-3x +3y -12z = -6

3x - 6y + 2z = 26

______________

-3y - 10z = 20 (5)

3. Solve the system created by equations (4) and (5).

3y + z = 7 

-3y -10z = 20 (add 2 equations)

____________

-9z = 27

z = -3

4. Now, substitute z = 3 into equation (4) to find y.

3y + z = 7    z = -3

3y -3 = 7

3y = 10 

y = 10/3

5. Use the answers and substitute into any equation involving the remaining variable (x).

x + 2y + 5z = 9  z = -3    y = 10/3

x + 20/3 -15 = 9

x = 52/3

6. Check the solution in all three original equations

1. 52/3 + 20/3 - 15 = 9

   72/3 = 24

   24 = 24

(2) x — y + 4z = 2

   52/3 - 10/3 - 12 = 2

   42/3 = 14

   14 = 14

(3) 3x — 6y + 2z = 26

  156/3 - 60/3 - 6 = 26

  96/3 = 32

  32 = 32

Give an example, with justification, of each of

the following : 2

1. An element of (C x Q) \(Q x C); 

(ii) An infinite set whose complement in R

is infinite. 

Answer: (1,3)

1. Let’s say set C = {1,2} and set Q = {3,4} 

   C x Q is a Cartesian product and can be constructed by associating every element of one set with every element of another set, so C x Q = {(1,3), (1,4), (2,3), (2,4)}

      Q x C can be constructed in the same way: Q x C = {(3,1), (3,2), (4,1), (4,2)}

(C x Q) \(Q x C) is relative complement of C x Q in Q x C is the set of all elements that are members of C x Q but not members of Q x C

We can see that all elements of C x Q are different from elements of Q x C (relative pair (1,3) is not the same as (3,1), the order is important), so (C x Q) \(Q x C) = C x Q. ((1,3), (1,4), (2,3), (2,4) are members of C x Q but not members of Q x C)

So the answer can be any element of C x D, let’s say (1,3).

(ii) An infinite set whose complement in R

is infinite.

Answer: We know that The natural numbers ℕ are contained in the integers ℤ, which are contained in the rational numbers ℚ, which are contained in the real numbers ℝ, which are contained in the complex numbers ℂ. And we know that all of those sets are infinite. So set of complex numbers C is the answer. It is infinite and it’s complement in R (all elements of C that are not in R) is also infinite: 3+2i, 1+i, 39+3i, …

2. (a) Find the roots of the polynomial equation

x⁴- 9x³+17x²+ 33x - 90 = 0

given that two of its roots are equal to 3

As we know 2 of 4 roots are equal to 3, so the 2 factors are (x-3).

So we need to find remaining quadratic equation and solve it to get 2 more roots.

x⁴- 9x³+17x²+ 33x - 90 = 0

x⁴- 9x³+17x²+ 33x - 90 = (x-a)(x-b)(x-3)(x-3)

Let’s divide our equation by (x-3)(x-3) = x² - 6x + 9

We got x² - 3x -10. Let’s solve it using factoring method. 

x² - 3x -10 = (x - 5)(x + 2)

x= 5, x= -2

Answer: x= 3, x= 3, x= 5, x= -2.

(b) Use the principle of mathematical induction

to show that for any positive integer n,

(311— 2n) > O. 

1. We need to prove that (311— 2n) > O is true for integer n=k then it will also be true for its successor, n = k + 1

We assume that (311 - 2n) > 0 is true for n=k, so (311 - 2k) >0

We must show that (311 - 2k)>0 is true for n=k+1

(311 - 2(k+1))>0

to do that we add the next term (k+1) to both sides

(311 -2(k+1)) + (k+1) > (k+1)

(311 - k - 1- k -1) >0

(311 - 2(k+1))>0

when -2k > -309

k < 154.5

n = k +1

so n < 155.5

2. Next we need to show that (311— 2n) > O is true for n=1 

311-2>0

309>0 which is true

We have now fulfilled both conditions of the principle of mathematical induction. The inequality is therefore true for every positive integer, when n < 155.5

(b) Use Cardano's method to obtain the roots of x³ - 3x + 2 = 0

Let's check the discriminant  Δ = – 4p³ – 27q² = 0

So the equation has a repeated root and all its roots are real

We don't need to depress the cubic equation because we don't have x² term.

Our equation has the form of y³ + Ay = B

First we substitute 3st = A and s³ - t³ = B and we solve for y= s - t

x³- 3x = -2

3st = -3

s³ - t³ = -2

s = -1/t

(-1/t)³ - t³ = -2 we multiply by t³

-t⁶ + 2t³ - 1 = 0

t³ = 1 we only chose real roots, because we know that our cubic equation has 3 real roots

t = 1

s³ - 1 = -2

s³ = -1

s = -1

Now we solwe for y = s -t

y= - 1 - 1 = -2

Our first root is x = -2

We divide x³ - 3x +2 by (x+2) and we get x² - 2x +1

We can use factoring to find our roots x² - 2x +1 = (x-1)(x-1)

Answer: x= -2, x= 1, x=1