Comprehensive Study Notes on Algebraic Foundations, Collecting Like Terms, and Polynomial Operations

Fundamentals of Algebraic Terminology and Expressions

An algebraic expression is built from terms, coefficients, and variables:

• Term: A single mathematical entity, which can be a constant number, a single variable, or a product of numbers and variables.

• Coefficient: The numerical factor multiplied by the variable or variables in a term.

• Variable: A symbol, typically a letter, used to represent an unknown or changing quantity.

Identification of Term Elements

• Term: $2y$

  • Coefficient: $2$

  • Variable: $y$

• Term: $-3x$

  • Coefficient: $-3$

  • Variable: $x$

• Term: $mn$

  • Coefficient: $1$

  • Variable: $mn$

• Term: $\frac{1}{2}x^2$

  • Coefficient: $\frac{1}{2}$

  • Variable: $x^2$

• Term: $-w^2$

  • Coefficient: $-1$

  • Variable: $w^2$

• Term: $-0.4gh^3$

  • Coefficient: $-0.4$

  • Variable: $gh^3$

Translating Verbal Statements into Algebraic Expressions

• Triple a number: $3n$

• Four more than a number: $n + 4$

• 5 less than double a number: $2n - 5$

• A number $x$ is tripled and then $19$ is subtracted from the result: $3x - 19$

• The variable $y$ is squared and then the result is increased by $10$: $y^2 + 10$

• The variable $n$ is decreased by $6$ and then the result is multiplied by $-8$: $-8(n - 6)$

• A number $p$ is increased by $70$ and then the result is divided by $6$: $\frac{p + 70}{6}$

Counting Terms in Algebraic Expressions

• $4x + 3$ contains $2$ terms ($4x$ and $3$).

• $y$ contains $1$ term ($y$).

• $6x^2 - 7x + 12$ contains $3$ terms ($6x^2$, $-7x$, and $12$).

• $-7m$ contains $1$ term ($-7m$).

• $8x^2 y^5 z^3$ contains $1$ term ($8x^2 y^5 z^3$).

• $9x^5 - 8xy + 4y^3 + 6$ contains $4$ terms ($9x^5$, $-8xy$, $4y^3$, and $6$).

Evaluating Expressions and Word Problems

Evaluating Expressions for Specific Variable Values

• Expression: $-5x + 12$ for $x = -3$

  • Substitution and Calculation: $-5(-3) + 12 = 15 + 12 = 27$

• Expression: $-8 + 20k$ for $k = -4$

  • Substitution and Calculation: $-8 + 20(-4) = -8 - 80 = -88$

• Expression: $7x^2 - 1$ for $x = 3$

  • Substitution and Calculation: $7(3)^2 - 1 = 7(9) - 1 = 63 - 1 = 62$

• Expression: $-3x^2 + 6x - 2$ for $x = 2$

  • Substitution and Calculation: $-3(2)^2 + 6(2) - 2 = -3(4) + 12 - 2 = -12 + 12 - 2 = -2$

• Expression: $\frac{t - 24}{3}$ for $t = 6$

  • Substitution and Calculation: $\frac{6 - 24}{3} = \frac{-18}{3} = -6$

• Expression: $2xy$ for $x = -2$ and $y = 3$

  • Substitution and Calculation: $2(-2)(3) = -12$

• Expression: $\frac{3}{5}x + 7$ for $x = 10$

  • Substitution and Calculation: $\frac{3}{5}(10) + 7 = 3(2) + 7 = 6 + 7 = 13$

• Expression: $-3a^2 + 4ab - 2b^2$ for $a = 0$ and $b = -1$

  • Substitution and Calculation: $-3(0)^2 + 4(0)(-1) - 2(-1)^2 = 0 + 0 - 2(1) = -2$

Fitness Club Earnings Application

Meredith works at a fitness club during the summer. She earns a bonus of $\$5$ for each student membership sold and $\$7$ for each adult membership sold.

• Total Bonus Expression: Let $s$ represent the number of student memberships and $a$ represent the number of adult memberships. The total bonus expression is $5s + 7a$

• Bonus Calculation: For $12$ student memberships and $10$ adult memberships:     $\text{Total Bonus} = 5(12) + 7(10) = 60 + 70 = 130$     Meredith earns a total bonus of $\$130$

Unit 2 Lesson 2: Collecting Like Terms

Definition and Classification of Terms

• Like terms are terms that contain identical variable factors raised to the exact same powers. Only their numerical coefficients may differ.

• Unlike terms have different variables or the same variables raised to different exponents.

Identification of Polynomials Without Like Terms

• A) $2x + 5 - 3x + 2xy$: Contains like terms $2x$ and $-3x$

• B) $3x^2 + 3xy + 3$: Contains no like terms. Every term has distinct variable powers and combinations.

• C) $4 - 9x + 9y + 3$: Contains like terms $4$ and $3$

• D) $-4a^3 + 5b - 2a^2 + 7b$: Contains like terms $5b$ and $7b$

Classifying Pairs of Terms as Like or Unlike

• a) $2x$ and $-5x$: Like

• b) $3y$ and $3z$: Unlike

• c) $-x^2$ and $\frac{1}{2}x^2$: Like

• d) $4a^2$ and $3a^3$: Unlike

• e) $2ab$ and $3a^2$: Unlike

• f) $5x^2 y$ and $-2xy^2$: Unlike

• g) $3uv$ and $2vu$: Like (commutative property applies: $uv = vu$)

• h) $9p^2 q^3$ and $-4q^3 p^2$: Like (commutative property applies)

Simplifying Expressions by Collecting Like Terms

• $3x + 5 + 2x + 1 = (3x + 2x) + (5 + 1) = 5x + 6$

• $2k + 3m + 4m + 6k = (2k + 6k) + (3m + 4m) = 8k + 7m$

• $3k - 5 + 8 - k + 1 - 4k = (3k - k - 4k) + (-5 + 8 + 1) = -2k + 4$

• $-6vw - 4v^2 w + 2v^2 w - 8wv = (-4v^2 w + 2v^2 w) + (-6vw - 8vw) = -2v^2 w - 14vw$

• $-2m + n - (-6m) - 3n - 4m + 5 = -2m + n + 6m - 3n - 4m + 5 = (-2m + 6m - 4m) + (n - 3n) + 5 = -2n + 5$

Advanced Multi-Variable Simplification

• $2x^2 y + 3xy - 7xy + 3x^2 y = (2 + 3)x^2 y + (3 - 7)xy = 5x^2 y - 4xy$

• $6xy^2 + 4x^2 y - 2xy^2 - 3x^2 y = (4 - 3)x^2 y + (6 - 2)xy^2 = x^2 y + 4xy^2$

• $-7a^2 b - 4ab^2 + 8ab^2 + 3a^2 b^2 - 2a^2 b = 3a^2 b^2 + (-7 - 2)a^2 b + (-4 + 8)ab^2 = 3a^2 b^2 - 9a^2 b + 4ab^2$

• $6xy + 6x^2 + 10yx + 2y^2 - 4xy = 6x^2 + (6 + 10 - 4)xy + 2y^2 = 6x^2 + 12xy + 2y^2$

Simplifying and Evaluating Expressions

For $x = -2$ and $y = 3$:

• Expression a: $8 + y - 3y^2 + 12 - 7y^2$

  • Simplified Form: $-10y^2 + y + 20$

  • Evaluation: $-10(3)^2 + 3 + 20 = -10(9) + 23 = -90 + 23 = -67$

• Expression b: $3x^2 - 4y + 7x - 8y^2 + 6y - 7x$

  • Simplified Form: $3x^2 - 8y^2 + 2y$

  • Evaluation: $3(-2)^2 - 8(3)^2 + 2(3) = 3(4) - 8(9) + 6 = 12 - 72 + 6 = -54$

Testing Algebraic Equivalence

• $5x + 3x$ and $8x$:     $5x + 3x = 8x$. These expressions are Equivalent.

• $7x - 3x$ and $3x - 7x$:     $7x - 3x = 4x$, but $3x - 7x = -4x$. These expressions are Not Equivalent.

• $3x^2 + x$ and $3x + x^2$:     The terms have different coefficients for corresponding powers of $x$. These expressions are Not Equivalent.

Unit 2 Lesson 3: Adding and Subtracting Polynomials

Basic Polynomial Addition and Subtraction

• Multiple Choice 1: $(2x - 7) + (3x + 8)$

  • Simplification: $(2x + 3x) + (-7 + 8) = 5x + 1$

  • Correct Option: C) $5x + 1$

• Exercise 2a: $(3x + 4) + (7x + 5) = (3x + 7x) + (4 + 5) = 10x + 9$

• Exercise 2b: $(y + 2) + (3 + 6y) = (y + 6y) + (2 + 3) = 7y + 5$

• Exercise 2c: $(5h + 9) - (-5h + 6) = 5h + 9 + 5h - 6 = (5h + 5h) + (9 - 6) = 10h + 3$

Multi-Term Polynomial Operations

• $(7x - 9) + (x - 4) = 8x - 13$

• $(8c - 6) - (c + 7) = 8c - 6 - c - 7 = 7c - 13$

• $(3p^2 - 8p + 1) + (9p^2 + 4p - 1) = (3p^2 + 9p^2) + (-8p + 4p) + (1 - 1) = 12p^2 - 4p$

• $(5xy^2 + 6x - 7y) - (3xy^2 - 6x + 7y) = 5xy^2 + 6x - 7y - 3xy^2 + 6x - 7y = 2xy^2 + 12x - 14y$

• $(4x - 3) + (x + 8) - (2x - 5) = 4x - 3 + x + 8 - 2x + 5 = 3x + 10$

• $(2uv^2 - 3v) - (v + 3u) + (4uv^2 - 9u) = 2uv^2 - 3v - v - 3u + 4uv^2 - 9u = 6uv^2 - 12u - 4v$

Simplification and Evaluation ($x = 2$, $y = -3$)

• Exercise 4a: $(7x + 3) - (-2x - 4)$

  • Simplified Form: $7x + 3 + 2x + 4 = 9x + 7$

  • Evaluation: $9(2) + 7 = 18 + 7 = 25$

• Exercise 4b: $17y^2 - (15y^2 + 3y - 8)$

  • Simplified Form: $17y^2 - 15y^2 - 3y + 8 = 2y^2 - 3y + 8$

  • Evaluation: $2(-3)^2 - 3(-3) + 8 = 2(9) + 9 + 8 = 18 + 9 + 8 = 35$

• Exercise 4c: $\left(\frac{1}{2}x^2 + \frac{2}{3}x - \frac{5}{3}\right) - \left(\frac{3}{8}x^2 - \frac{3}{4}x + \frac{7}{3}\right)$

  • Distribute negative sign: $\frac{1}{2}x^2 + \frac{2}{3}x - \frac{5}{3} - \frac{3}{8}x^2 + \frac{3}{4}x - \frac{7}{3}$

  • Combine $x^2$ terms: $\left(\frac{4}{8} - \frac{3}{8}\right)x^2 = \frac{1}{8}x^2$

  • Combine $x$ terms: $\left(\frac{8}{12} + \frac{9}{12}\right)x = \frac{17}{12}x$

  • Combine constant terms: $-\frac{5}{3} - \frac{7}{3} = -\frac{12}{3} = -4$

  • Simplified Form: $\frac{1}{8}x^2 + \frac{17}{12}x - 4$

  • Evaluation for $x = 2$:         $\frac{1}{8}(2)^2 + \frac{17}{12}(2) - 4 = \frac{4}{8} + \frac{34}{12} - 4 = \frac{1}{2} + \frac{17}{6} - 4 = \frac{3}{6} + \frac{17}{6} - \frac{24}{6} = -\frac{4}{6} = -\frac{2}{3}$

Geometric Applications and Perimeter Calculations

Rectangular Field Dimensions

For a rectangular field whose length is three times its width $w$:

• Expression for Perimeter:     $\text{Perimeter} = 2(\text{length}) + 2(\text{width}) = 2(3w) + 2(w) = 6w + 2w = 8w$

• Perimeter for Width $w = 300\,\text{m}$:     $\text{Perimeter} = 8(300) = 2400\,\text{m}$

• Dimensions for Perimeter $P = 1600\,\text{m}$:     $8w = 1600 \implies w = 200\,\text{m}$     $\text{Length} = 3(200) = 600\,\text{m}$

Trapezoid Perimeter Calculation

• Side lengths: Top = $x$, Left = $2y$, Bottom = $3x$, Right = $4y$

• Perimeter Calculation:     $\text{Perimeter} = x + 2y + 3x + 4y = (x + 3x) + (2y + 4y) = 4x + 6y$

Swimming Pool Installation

A swimming pool manufacturer installs rectangular pools whose length is twice the width plus $5\,\text{m}$.

• Dimensions: Width = $w$, Length = $2w + 5$

• Perimeter Expression:     $\text{Perimeter} = 2(\text{Length}) + 2(\text{Width}) = 2(2w + 5) + 2w = 4w + 10 + 2w = 6w + 10$

• Perimeter for Width $w = 6\,\text{m}$:     $\text{Perimeter} = 6(6) + 10 = 36 + 10 = 46\,\text{m}$

Rectangle Perimeter Diagram

• Side lengths: Length = $2x + 5$, Width = $x - 4$

• Perimeter Expression:     $\text{Perimeter} = 2(2x + 5) + 2(x - 4) = 4x + 10 + 2x - 8 = 6x + 2$

Triangle Missing Side Determination

For a triangle with a perimeter of $2x^2 + x + 13$ and two known side lengths:

• Side 1: $2x^2 - 4x + 8$

• Side 2: $-x^2 + 2x + 1$

• Sum of known sides:     $\text{Side}_1 + \text{Side}_2 = (2x^2 - 4x + 8) + (-x^2 + 2x + 1) = x^2 - 2x + 9$

• Length of the third side:     $\text{Third Side} = \text{Perimeter} - (\text{Side}_1 + \text{Side}_2)$     $\text{Third Side} = (2x^2 + x + 13) - (x^2 - 2x + 9) = 2x^2 + x + 13 - x^2 + 2x - 9 = x^2 + 3x + 4$