Complex Numbers and Factoring Polynomials

Complex Numbers: The Imaginary Unit and Standard Form

Imaginary Number (i):
- Defined as the square root of negative one: $i = \sqrt{-1}$ .
- Allows for working with imaginary and complex numbers.
Complex Number (Standard Form):
- Expressed as $a + bi$ , where $a$ and $b$ are real numbers.
- A $is the real part.</li><li>B$ is the imaginary part.
Properties of Complex Numbers (for $a + bi$ and $c + di$ in standard form):
- Addition: Add real parts and imaginary parts separately.
  - $(a + bi) + (c + di) = (a + c) + (b + d)i$
  - Example: $(-2 - 4i) + (5 + 2i) = (-2 + 5) + (-4 + 2)i = 3 - 2i$
- Subtraction: Subtract real parts and imaginary parts separately.
  - $(a + bi) - (c + di) = (a - c) + (b - d)i$
  - Example: $(3 - i) - (7 - 9i) = (3 - 7) + (-1 - (-9))i = -4 + 8i$
- Multiplication: Follows the FOIL (First, Outer, Inner, Last) method, similar to binomials.
  - $(a + bi)(c + di) = ac + adi + bci + bdi^2$
  - Key property: $i^2 = (\sqrt{-1})^2 = -1$
  - Therefore, $(a + bi)(c + di) = ac + adi + bci - bd = (ac - bd) + (ad + bc)i$
  - Example 1: $(-2 + 6i)(4 - 3i)$
    - Using FOIL: $(-2)(4) + (-2)(-3i) + (6i)(4) + (6i)(-3i)$
    - $-8 + 6i + 24i - 18i^2$
    - $-8 + 30i - 18(-1) = -8 + 30i + 18 = 10 + 30i$
  - Example 2: $(5 + 2i)(5 - 2i)$
    - Using FOIL: $(5)(5) + (5)(-2i) + (2i)(5) + (2i)(-2i)$
    - $25 - 10i + 10i - 4i^2$
    - $25 - 4(-1) = 25 + 4 = 29$

Complex Conjugates

Definition:
- For a complex number $a + bi$ , its complex conjugate is $a - bi$ .
- The sign of the imaginary part is flipped.
Property of Multiplication:
- When a complex number is multiplied by its complex conjugate, the result is always a real number.
- $(a + bi)(a - bi) = a^2 - abi + abi - b^2i^2 = a^2 - b^2(-1) = a^2 + b^2$
Cautionary Note: This differs from real conjugates (used for rationalizing denominators with square roots), where $(a + b)(a - b) = a^2 - b^2$ . The presence of $i^2 = -1$ in complex conjugates changes the sign to a sum of squares.

Dividing Complex Numbers

General Process:
1. Multiply both the numerator and the denominator by the complex conjugate of the denominator.
2. Simplify the expression.
3. Write the final answer in standard form ( $a + bi$ ).
Example 1: $-2 / (5i)$
1. Identify complex conjugate of denominator ( $5i$ ): $-5i$ .
2. Multiply: $(-2 / (5i)) \cdot (-5i / (-5i)) = 10i / (-25i^2)$
3. Simplify using $i^2 = -1$ : $10i / (25) = (10/25)i = (2/5)i$
4. Standard form: $0 + (2/5)i$ (The instructor mentioned writing it as $(2/5)i$ or $2i/5$ is acceptable, implying $a=0$ in this case, but warned against leaving it as $(3-2i)/5$ for other examples, as it's not standard form until separated).
Example 2: $(8 + 2i) / (3 - 5i)$
1. Identify complex conjugate of denominator ( $3 - 5i$ ): $3 + 5i$ .
2. Multiply:
  - Numerator: $(8 + 2i)(3 + 5i) = (8)(3) + (8)(5i) + (2i)(3) + (2i)(5i)$
    - $24 + 40i + 6i + 10i^2 = 24 + 46i - 10 = 14 + 46i$
  - Denominator: $(3 - 5i)(3 + 5i) = 3^2 + 5^2 = 9 + 25 = 34$
3. Combine: $(14 + 46i) / 34$
4. Write in standard form and simplify (divide each term by $34$ ):
  - $(14/34) + (46/34)i = (7/17) + (23/17)i$

Factoring Polynomials: Introduction

Definition: The process of rewriting a polynomial as a product of other polynomials (its factors).
Importance: Simplifies solving polynomial equations.
- Example: Solving $x^2 - 19x + 88 = 0$ is easier when factored to $(x - 8)(x - 11) = 0$ , leading to solutions $x = 8$ or $x = 11$ .
General First Steps for Factoring (Always Apply):
1. Pull out the Greatest Common Factor (GCF) from all terms.
  - The GCF is the largest factor shared by all terms.
2. Ensure the Leading Term has a Positive Coefficient.
  - If the term with the highest power of the variable has a negative coefficient, factor out a negative sign (part of the GCF if applicable).
GCF Practice Examples:
- Example 1: $12x^5 + 18x^4 - 24x^3$
  - Coefficients ( $12, 18, 24$ ) GCF: $6$
  - Variables ( $x^5, x^4, x^3$ ) GCF (lowest power): $x^3$
  - Factored result: $6x^3(2x^2 + 3x - 4)$
- Example 2: $-4x^2 - 8x + 12$
  - Coefficients ( $4, 8, 12$ ) GCF: $4$
  - No common variable (constant term $12$ has no $x$ ).
  - Leading term is negative, so pull out $-4$ .
  - Factored result: $-4(x^2 + 2x - 3)$
- Example 3 (Recognizing a binomial as a GCF): $3y(2y - 5) + (2y - 5)$
  - The GCF is the binomial $(2y - 5)$ .
  - Think of the second term as $1 \cdot (2y - 5)$ .
  - Factored result: $(2y - 5)(3y + 1)$

Factoring by Grouping (for Four Terms)

Applicability: Used for polynomials with four terms.
Steps:
1. (Implicit: Apply general first steps - GCF and positive leading term).
2. Group terms into pairs (e.g., first two and last two).
3. Factor out the GCF from each pair.
4. Factor out the shared binomial term (if it exists).
- Note: Sometimes, terms might need to be rearranged to find appropriate pairs for grouping.
Example 1: $6ab - 21a + 4b - 14$
1. No common GCF for all terms (other than $1$ ). Leading term ( $6ab$ ) is positive.
2. Group: $(6ab - 21a) + (4b - 14)$
3. Factor GCF from each pair:
  - $3a(2b - 7)$
  - $+2(2b - 7)$
4. Factor out shared binomial $(2b - 7)$ :
  - $(2b - 7)(3a + 2)$
Example 2 (Requiring Rearrangement): $3ry + 2s + sy + 6r$
1. No common GCF. Leading term ( $3ry$ ) is positive.
2. Initial grouping ( $(3ry + 2s)$ and $(sy + 6r)$ ) yields no common factors within pairs.
3. Rearrange terms to group common factors (e.g., switch $2s$ and $sy$ ):
  - $3ry + sy + 2s + 6r$
4. New grouping: $(3ry + sy) + (2s + 6r)$
5. Factor GCF from each pair:
  - $y(3r + s)$
  - $+2(s + 3r)$ (Note: $(s + 3r)$ is the same as $(3r + s)$ .)
6. Factor out shared binomial $(3r + s)$ :
  - $(3r + s)(y + 2)$

Factoring Trinomials (for Three Terms)

General Form: $ax^2 + bx + c$ (where $a \neq 0$ )

Method 1: Inspection (for $a=1$ )

Applicability: When the coefficient of the squared term ( $a$ ) is $1$ (e.g., $x^2 + bx + c$ ).
Process: Find two numbers, $u$ and $v$ , such that:
- $u \cdot v = c$ (the constant term)
- $u + v = b$ (the coefficient of the linear term)
Then, the factored form is $(x + u)(x + v)$ .
Example 1: $x^2 - 10x + 16$
- $c = 16$ (product)
- $b = -10$ (sum)
- Numbers ( $u, v$ ): $-8$ and $-2$ ( $(-8) \cdot (-2) = 16$ , $-8 + (-2) = -10$ )
- Factored form: $(x - 8)(x - 2)$
Example 2: $z^2 + 6z - 27$
- $c = -27$ (product)
- $b = 6$ (sum)
- Numbers ( $u, v$ ): $9$ and $-3$ ( $(9) \cdot (-3) = -27$ , $9 + (-3) = 6$ )
- Factored form: $(z + 9)(z - 3)$

Method 2: The AC Method (for any $a \neq 0$ )

Steps:
1. (Implicit: Apply general first steps - GCF and positive leading term).
2. Determine the product $AC$ and the coefficient B$.
 - AC $is the product of the coefficient of the squared term and the constant term.</li><li>$ B $is the coefficient of the linear term.</li></ul></li><li>Find two numbers,$ u $and$ v $, such that:<ul><li>$ u \cdot v = AC $</li><li>$ u + v = B $</li></ul></li><li>Rewrite the problem: Replace the middle term$ Bx $with$ Ux + Vx $(using the$ u $and$ v $found).</li><li>Factor by Grouping (now a 4-term polynomial).</li></ol></li><li>Example 1:$ 12x^2 - 5x - 2 $<ol><li>No GCF. Leading term ($ 12x^2 $) positive.</li><li>$ AC = (12)(-2) = -24 $.$ B = -5 $.</li><li>Find$ u, v $:$ -8 $and$ 3 $($ (-8) \cdot 3 = -24 $,$ -8 + 3 = -5 $).</li><li>Rewrite:$ 12x^2 - 8x + 3x - 2 $</li><li>Factor by Grouping:<ul><li>Initial grouping of$ (12x^2 - 8x) $and$ (3x - 2) $works.</li><li>$ 4x(3x - 2) + 1(3x - 2) $</li><li>Final form:$ (3x - 2)(4x + 1) $</li><li>(Self-correction during lecture): If initial grouping was$ 12x^2 + 3x - 8x - 2 $, then:$ 3x(4x + 1) - 2(4x + 1) = (4x + 1)(3x - 2) $. The order of$ ux $and$ vx $terms can sometimes affect the ease of grouping, but the final result will be the same.</li></ul></li></ol></li><li>Example 2 (with GCF and multiple variables):$ -20c^3 + 34c^2d - 6cd^2 $<ol><li>GCF and Positive Leading Term: GCF for$ 20, 34, 6 $is$ 2 $. All terms have at least one$ c $. Leading term is negative. So, pull out$ -2c $.<ul><li>$ -2c(10c^2 - 17cd + 3d^2) $</li></ul></li><li>Determine$ AC $and$ B $ (for the trinomial$ 10c^2 - 17cd + 3d^2 $):<ul><li>$ AC = (10)(3) = 30 $.</li><li>$ B = -17 $.</li></ul></li><li>Find$ u, v $:$ -15 $and$ -2 $($ (-15) \cdot (-2) = 30 $,$ -15 + (-2) = -17 $).</li><li>Rewrite: (Don't forget the$ -2c $GCF factored out earlier).<ul><li>$ -2c(10c^2 - 2cd - 15cd + 3d^2) $</li></ul></li><li>Factor by Grouping:<ul><li>$ -2c[ (10c^2 - 2cd) + (-15cd + 3d^2) ] $</li><li>$ -2c[ 2c(5c - d) - 3d(5c - d) ] $</li><li>Factor out shared binomial$ (5c - d) $:$ -2c(5c - d)(2c - 3d) $</li></ul></li></ol></li></ul><h5 id="92ed474f-b5f1-496d-84e0-2031334fd7b6" data-toc-id="92ed474f-b5f1-496d-84e0-2031334fd7b6" collapsed="false" seolevelmigrated="true">Special Factoring Forms</h5><h6 id="25fb40dc-f8e7-437c-a1b9-1e39ad4cde94" data-toc-id="25fb40dc-f8e7-437c-a1b9-1e39ad4cde94" collapsed="false" seolevelmigrated="true">Difference of Squares</h6><ul><li>Form:$ a^2 - b^2 $</li><li>Factored Form:$ (a + b)(a - b) $</li><li>Recognition: Look for two perfect squares separated by a minus sign.</li><li>Example 1:$ 16x^2 - 9 $<ul><li>Recognize$ 16x^2 = (4x)^2 $and$ 9 = 3^2 $.</li><li>Factored form:$ (4x + 3)(4x - 3) $</li></ul></li><li>Example 2 (Multi-step):$ z^4 - 81 $<ol><li>Recognize$ z^4 = (z^2)^2 $and$ 81 = 9^2 $.</li><li>Initial factoring:$ (z^2 + 9)(z^2 - 9) $</li><li>Notice that$ (z^2 - 9) $is another difference of squares ($ z^2 - 3^2 $).</li><li>Factor the second term further:$ (z^2 + 9)(z + 3)(z - 3)$$
- This is the fully factored form.

Complex Numbers and Factoring Polynomials

Complex Numbers: The Imaginary Unit and Standard Form

Complex Conjugates

Dividing Complex Numbers

Factoring Polynomials: Introduction

Factoring by Grouping (for Four Terms)

Factoring Trinomials (for Three Terms)

Method 1: Inspection (for a=1a=1a=1)

Method 2: The AC Method (for any a≠0a \neq 0a=0)

Method 1: Inspection (for $a=1$ )

Method 2: The AC Method (for any $a \neq 0$ )