Math Problem Statement

Math_1319_Quiz 2.pdf

39.41 KB

Solution

Let's go step-by-step to solve each question in your quiz:


1. Evaluate the following:

(a) (24i)(4+3i)(2 - 4i) - (-4 + 3i)
Distribute the negative sign:
(24i)+43i(2 - 4i) + 4 - 3i
Now combine like terms:
2+4=62 + 4 = 6
4i3i=7i-4i - 3i = -7i
The result is:
67i\boxed{6 - 7i}

(b) i8i^8
Recall that powers of ii follow a cycle:
i1=i,i2=1,i3=i,i4=1,and the cycle repeats.i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and the cycle repeats.}
Since 8÷4=28 \div 4 = 2 with no remainder, i8i^8 corresponds to i4=1i^4 = 1.
Thus:
1\boxed{1}


2. Let f(x)=x2(x3)2(x5)(x+3)f(x) = x^2 (x - 3)^2 (x - 5) (x + 3). Find the solution set (in interval notation) for f(x)<0f(x) < 0.

  1. Identify the roots:

    • x=0,3,5,3x = 0, 3, 5, -3.
  2. Analyze the sign changes around these points.

    • Between the roots, the factors x2x^2, (x3)2(x - 3)^2, (x5)(x - 5), and (x+3)(x + 3) may switch sign.
  3. Since x2x^2 and (x3)2(x - 3)^2 are always non-negative, the sign changes only depend on (x5)(x - 5) and (x+3)(x + 3).

  4. Test intervals between the roots:

    • x<3x < -3: All factors negative, so f(x)>0f(x) > 0.
    • 3<x<0-3 < x < 0: Positive from x2x^2 and (x+3)(x + 3), negative from others, so f(x)<0f(x) < 0.
    • 0<x<30 < x < 3: All factors positive, f(x)>0f(x) > 0.
    • 3<x<53 < x < 5: Positive except (x5)(x - 5), so f(x)<0f(x) < 0.
    • x>5x > 5: All factors positive, f(x)>0f(x) > 0.
  5. Solution:
    f(x)<0f(x) < 0 on (3,0)(3,5)(-3, 0) \cup (3, 5).

(3,0)(3,5)\boxed{(-3, 0) \cup (3, 5)}


3. Solve 4x2=3x4x^2 = 3x by factoring.

  1. Rearrange the equation: 4x23x=04x^2 - 3x = 0

  2. Factor out xx: x(4x3)=0x(4x - 3) = 0

  3. Set each factor equal to zero:

    • x=0x = 0
    • 4x3=04x=3x=344x - 3 = 0 \quad \Rightarrow \quad 4x = 3 \quad \Rightarrow \quad x = \frac{3}{4}
  4. Solution: x=0 or x=34\boxed{x = 0 \text{ or } x = \frac{3}{4}}


Would you like any part of the solutions explained in more detail?
Here are five follow-up questions to explore:

  1. How can I plot complex numbers on the complex plane?
  2. What’s the general solution pattern for inequalities involving polynomials?
  3. How do cycles of ii apply to more complicated exponents?
  4. What’s another way to solve quadratic equations like 4x23x=04x^2 - 3x = 0?
  5. How do we know when factoring will work versus other methods like the quadratic formula?

Tip: Always analyze roots and their multiplicities when finding where a polynomial is positive or negative. This helps you determine if a root "bounces" or crosses the x-axis.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Complex Numbers
Polynomial Inequalities
Quadratic Equations

Formulas

Complex number subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Powers of i: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1
Factoring: ax^2 + bx + c = 0

Theorems

Fundamental Theorem of Algebra
Properties of powers of imaginary unit i
Sign analysis for polynomial functions

Suitable Grade Level

Grades 10-12