Solve the equation

Question

1. Solve the equation 2x^3 + 3x^2 - 8x + 3 = 0 

2. Let vector a = (2,5) and b = (1, -1). Find the value of the constant t which minimizes |a + tb|. 

3. Find the angle a between the two lines x - 2y = 3 and 3x - y = 2, where the range of values of a is 0<= a <= phi/2 

4. Let a and b be non-real roots of the equation x^3 = 8. Find the value of a^2 + b^2 

5. Let f(x) = cos x and g(x) = sin x. Calculate lim h-0 f(x-2h)-f(x+h) / g(x+3h)-g(x-h)

Answer 1

1) The plonomial equation 2x³ + 3x²- 8x + 3 = 0

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

2x³ + 3x²- 8x + 3 = 0

If p/q is a rational zero, then p is a factor of 3 and q is a factor of 2.

The possible values of p are   ± 1 and  ± 3.

The possible values of q are ± 1 and ± 2 .

So, p/q = ± 1,  ± 3, ± 1/2 and ± 3/2 .

Make a table for the synthetic division and test possible  zeros.

p/q	2	3	-8	3
-1	2	1	-9	12
1	2	5	-3	0

Since f (1) =  0,  x = 1 is a zero. The depressed polynomial is   2x²+ 5x - 3

Since the depressed polynomial of this zero, 2x²+ 5x - 3, is quadratic, use the factorization to find the roots of the related quadratic equation 2x²+ 5x - 3 = 0.

2x²+ 6x - x - 3 = 0

2x(x + 3) -1 (x + 3) = 0

(x + 3) (2x - 1) = 0

Apply zero product property.

x + 3 = 0 and 2x - 1 = 0

x = -3 and 2x = 1

x = -3 and x = 1/2

So - 3 and 1/2 are also zeros of the function.

Solutions of the equtions are x = - 3, 1 and 1/2.

Answer 2

4) The equation

Apply the formula

Apply zero product property.

x = 2 is a real root of the equation

• Solve the equation

Compare it to quadratic form

Roots are

Non real roots of are

• 

Answer 3

3) The lines are x - 2y = 3 and 3x - y = 2

Write the two lines in slope intercept form y = mx + c.

Now the lines are y =  x/2 - 3/2 and y = 3x - 2

the lines slopes are

If the slopes of two lines are , the angle is obtained from

Fromula for angle between two lines is

Where

In this case

Substitute the above values in formula.

Angle between the lines x - 2y = 3 and 3x - y = 2 is 45 degrees.

Answer 4

5)

The solution is 3/4 tan(x ).

Answer 5

2) The vectors a (2, 5) and b  = (1, -1)

The vector equation of a line passing through the point a  and in the direction b  is a + tb

a +tb = (2,5) + t (1,-1)

= (2,5) + (t,-t)

= (2 + t, 5 - t)

Magnitude of a  vector is

Magnitude of a + tb is

To find the minimum value of |a +tb| 

• Take the derivative.
• Set it equals to 0, determine the critical point and it is the value of the constant t which minimizes

|a + tb |.