Math 424 Sample Test 1 Study Guide

Overview and General Information

• Course: Math 424
• Assessment: Sample Test Quarter 1 Test 1
• Core Topics Covered:
  • Complete polynomial factoring (grouping, difference of squares, sum/difference of cubes, trinomial factoring)
  • Polynomial multiplication and expansion with exponent variables
  • Symbolic logic, biconditional statements, and direct validity proofs
  • Algebraic inequalities over the integer domain $\mathbb{Z}$
  • Solving polynomial equations by expansion and reduction
  • Determining tautologies using logical laws and truth evaluations
  • Testing logical equivalence via De Morgan's Laws and truth tables
  • Evaluating compound truth values given atomic truth assignments
  • Structural analysis of sum and difference of cubes formulas
  • Conditional logic transformations: converse, inverse, and contrapositive

Polynomial Factoring and Algebraic Simplification

Question 1: Trinomial Factoring with Two Variables

• Problem Statement: Factor completely: $4x^2 - xy - 3y^2$
• Solution Method:
  • Identify the coefficients: $a = 4$, $b = -1$, $c = -3$.
  • Find two numbers that multiply to $a \times c = 4 \times (-3) = -12$ and add to $b = -1$.
  • The two numbers are $+3$ and $-4$.
  • Rewrite the middle term $-xy$ as $-4xy + 3xy$:     $4x^2 - 4xy + 3xy - 3y^2$
  • Factor by grouping:     $4x(x - y) + 3y(x - y) = (4x + 3y)(x - y)$
• Final Answer: $(4x + 3y)(x - y)$

Question 2: Four-Term Factoring by Grouping

• Problem Statement: Factor completely: $9x^2 - x^2 y^2 - 8y^2 + 72$
• Solution Method:
  • Rearrange terms to group common factors:     $(9x^2 + 72) - (x^2 y^2 + 8y^2)$
  • Factor out the greatest common factor (GCF) from each group:     $9(x^2 + 8) - y^2(x^2 + 8)$
  • Factor out the common binomial $(x^2 + 8)$:     $(9 - y^2)(x^2 + 8)$
  • Recognize that $(9 - y^2)$ is a difference of squares $a^2 - b^2 = (a - b)(a + b)$, where $a = 3$ and $b = y$:     $9 - y^2 = (3 - y)(3 + y)$
• Final Answer: $(3 - y)(3 + y)(x^2 + 8)$

Question 3: Difference of Squares with Quadratic Binomials

• Problem Statement: Factor completely: $(x^2 - 4x - 5)^2 - (x^2 + 6x + 5)^2$
• Solution Method:
  • Apply the difference of squares identity $A^2 - B^2 = (A - B)(A + B)$, where:     $A = x^2 - 4x - 5$$B = x^2 + 6x + 5$
  • Compute the difference $(A - B)$:     $A - B = (x^2 - 4x - 5) - (x^2 + 6x + 5) = x^2 - 4x - 5 - x^2 - 6x - 5 = -10x - 10 = -10(x + 1)$
  • Compute the sum $(A + B)$:     $A + B = (x^2 - 4x - 5) + (x^2 + 6x + 5) = 2x^2 + 2x = 2x(x + 1)$
  • Multiply $(A - B)$ and $(A + B)$ together:     $(A - B)(A + B) = [-10(x + 1)] \times [2x(x + 1)] = -20x(x + 1)^2$
• Final Answer: $-20x(x + 1)^2$

Question 4: Four-Term Cubic Factoring by Grouping

• Problem Statement: Factor completely: $a^3 + b^3 + a^2 b + ab^2$
• Solution Method:
  • Rearrange and group terms:     $(a^3 + a^2 b) + (ab^2 + b^3)$
  • Factor out the GCF from each group:     $a^2(a + b) + b^2(a + b)$
  • Factor out the common binomial $(a + b)$:     $(a^2 + b^2)(a + b)$
• Final Answer: $(a^2 + b^2)(a + b)$

Question 5: Polynomial Expansion with General Exponents

• Problem Statement: Write as a polynomial in simple form: $(x^n - 4y^n)(x^{2n} + 4x^n + 16y^{2n})$
• Solution Method:
  • Expand using the distributive property:     $(x^n - 4y^n)(x^{2n} + 4x^n + 16y^{2n}) = x^n(x^{2n} + 4x^n + 16y^{2n}) - 4y^n(x^{2n} + 4x^n + 16y^{2n})$
  • Multiply each term using exponent rules $x^a \times x^b = x^{a+b}$:     $x^{3n} + 4x^{2n} + 16x^n y^{2n} - 4x^{2n} y^n - 16x^n y^n - 64y^{3n}$
  • If the middle expression was intended as the difference of cubes identity $(A - B)(A^2 + AB + B^2) = A^3 - B^3$ with $A = x^n$ and $B = 4y^n$ (where the middle term is $4x^n y^n$):     $(x^n)^3 - (4y^n)^3 = x^{3n} - 64y^{3n}$
• Final Answer: $x^{3n} - 64y^{3n}$ (or expanded as $x^{3n} + 4x^{2n} - 4x^{2n}y^n + 16x^n y^{2n} - 16x^n y^n - 64y^{3n}$ as written)

Question 6: Difference of Squares Grouping Strategy

• Problem Statement: Write as a polynomial in simple form: $[a - 3b - 2y][a - 3b + 2y]$
• Solution Method:
  • Group the terms to form a difference of squares $[(a - 3b) - 2y][(a - 3b) + 2y]$:     $= (a - 3b)^2 - (2y)^2$
  • Expand $(a - 3b)^2$:     $(a - 3b)^2 = a^2 - 6ab + 9b^2$
  • Subtract $(2y)^2 = 4y^2$:     $a^2 - 6ab + 9b^2 - 4y^2$
• Final Answer: $a^2 - 6ab + 9b^2 - 4y^2$

Symbolic Logic, Truth Values, and Proofs

Question 7: Biconditional Property Verification

• Problem Statement: Write symbolically the following statement, then prove or disprove its validity: "If a biconditional is true and one of its statements is false, then the other statement must be false."
• Symbolic Representation:
  • Let the two component statements be $p$ and $q$
  • Biconditional statement: $p \rightarrow q$ and $q \rightarrow p$, represented as $p \nleftrightarrow q$
  • Premise 1: The biconditional is true: $(p \nleftrightarrow q)$
  • Premise 2: One statement is false: $\sim p$
  • Conclusion: The other statement is false: $\sim q$
  • Complete Symbolic Form: ((p \nleftrightarrow q) \nland \sim p) \rightarrow \sim q
• Proof / Disproof:
  • Validity: VALID (True)
  • Formal Proof via Truth Definition:
  • A biconditional statement $p \nleftrightarrow q$ evaluates to True if and only if both $p$ and $q$ share identical truth values (both are True, or both are False).
  • Given that $p \nleftrightarrow q$ is True, $p$ and $q$ must have the same truth value.
  • Given that one of the statements (say $p$) is False ($\sim p$ is True), the second statement $q$ cannot be True (which would yield $p \nleftrightarrow q \equiv \text{False}$).
  • Therefore, $q$ must be False ($\sim q$ is True).
  • Hence, the statement is proved to be VALID.

Algebraic Inequalities and Domain Restrictions

Question 8: Solving Linear-Quadratic Inequalities over Integers

• Problem Statement: Find the solution set for the following. The domain is $\mathbb{Z}$.   $4 - \frac{x}{2} - x(x - 5) > 3 + 6x - x^2$
• Step-by-Step Solution:
  • Expand the terms on the left-hand side:     $4 - \frac{x}{2} - x^2 + 5x > 3 + 6x - x^2$
  • Add $x^2$ to both sides to eliminate quadratic terms:     $4 - \frac{x}{2} + 5x > 3 + 6x$
  • Combine like terms on the left side ($-\frac{1}{2}x + 5x = \frac{9}{2}x$):     $4 + \frac{9}{2}x > 3 + 6x$
  • Subtract $3$ from both sides:     $1 + \frac{9}{2}x > 6x$
  • Subtract $\frac{9}{2}x$ from both sides:     $1 > 6x - \frac{9}{2}x$$1 > \frac{3}{2}x$
  • Multiply both sides by $\frac{2}{3}$:     $x < \frac{2}{3}$
  • Apply the domain restriction $\mathbb{Z}$ (set of all integers):
  • Integers strictly less than $\frac{2}{3}$ are $0, -1, -2, -3, \dots$
• Final Solution Set: $\{x \in \mathbb{Z} \mid x \le 0\}$ or $\{\dots, -3, -2, -1, 0\}$

Non-Linear Algebraic Equations

Question 9: Solving Cubic Polynomial Equations

• Problem Statement: Solve: $(x + 1)^3 = x^3 + 3x(x - 1)$
• Step-by-Step Solution:
  • Expand the left-hand side using the binomial cube formula $(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$:     $(x + 1)^3 = x^3 + 3x^2 + 3x + 1$
  • Expand the right-hand side:     $x^3 + 3x(x - 1) = x^3 + 3x^2 - 3x$
  • Set the expanded expressions equal to each other:     $x^3 + 3x^2 + 3x + 1 = x^3 + 3x^2 - 3x$
  • Subtract $x^3 + 3x^2$ from both sides:     $3x + 1 = -3x$
  • Add $3x$ to both sides:     $6x + 1 = 0$
  • Subtract $1$ and divide by $6$:     $6x = -1 \implies x = -\frac{1}{6}$
• Final Answer: $x = -\frac{1}{6}$

Tautologies and Logical Equivalence

Question 10: Tautology Evaluation 1

• Expression: $[(p \lor \sim r) \land \sim(\sim p)] \rightarrow \sim r$
• Analysis:
  • Simplify $\sim(\sim p)$ using double negation: $p$
  • Substitute back into the expression: $[(p \lor \sim r) \land p] \rightarrow \sim r$
  • Apply the Absorption Law $(p \lor A) \land p \equiv p$:     $p \rightarrow \sim r$
  • Test truth values: If $p = \text{True}$ and $r = \text{True}$, then $\sim r = \text{False}$.     $p \rightarrow \sim r \equiv \text{True} \rightarrow \text{False} \equiv \text{False}$
  • Since the statement is false in at least one case, it is not always true.
• Answer: NO

Question 11: Tautology Evaluation 2

• Expression: $(\sim t \land s) \rightarrow \sim t$
• Analysis:
  • Statement structure: $(A \land B) \rightarrow A$, where $A = \sim t$ and $B = s$
  • By the law of simplification, a conjunction $(A \land B)$ directly implies its individual components $A$ and $B$
  • If $(\sim t \land s)$ is True, then $\sim t$ MUST be True, making $\text{True} \rightarrow \text{True} \equiv \text{True}$
  • If $(\sim t \land s)$ is False, an implication with a false antecedent is vacuously True ($\text{False} \rightarrow \text{anything} \equiv \text{True}$)
  • Therefore, the statement is True for all possible truth assignments.
• Answer: YES

Question 12: Logical Equivalence Determination

• Statements to Compare: $\sim(p \land \sim r)$ and $\sim p \lor r$
• Proof of Equivalence:
  • Apply De Morgan's Law to the first statement $\sim(A \land B) \equiv \sim A \lor \sim B$:     $\sim(p \land \sim r) \equiv \sim p \lor \sim(\sim r)$
  • Apply the Law of Double Negation $\sim(\sim r) \equiv r$:     $\sim p \lor \sim(\sim r) \equiv \sim p \lor r$
  • Both statements yield identical truth tables across all 4 combinations of $p$ and $r$:
  • When $p=\text{T}, r=\text{T}$: $\sim(\text{T} \land \text{F}) = \text{T}$; $\text{F} \lor \text{T} = \text{T}$
  • When $p=\text{T}, r=\text{F}$: $\sim(\text{T} \land \text{T}) = \text{F}$; $\text{F} \lor \text{F} = \text{F}$
  • When $p=\text{F}, r=\text{T}$: $\sim(\text{F} \land \text{F}) = \text{T}$; $\text{T} \lor \text{T} = \text{T}$
  • When $p=\text{F}, r=\text{F}$: $\sim(\text{F} \land \text{T}) = \text{T}$; $\text{T} \lor \text{F} = \text{T}$
• Answer: YES, they are logically equivalent because applying De Morgan's Law and Double Negation transforms $\sim(p \land \sim r)$ directly into $\sim p \lor r$.

Truth Value Evaluation of Compound Propositions

Question 13: Given Truth Conditions

• Given True Statements:
  • $p = \text{True}$
  • $\sim q = \text{True} \implies q = \text{False}$
  • $\sim(\sim s) = \text{True} \implies s = \text{True}$
  • $r = \text{True}$
• Summary of Truth Values:
  • Atomic variables: $p = \text{True}$, $q = \text{False}$, $r = \text{True}$, $s = \text{True}$
  • Negations: $\sim p = \text{False}$, $\sim q = \text{True}$, $\sim r = \text{False}$, $\sim s = \text{False}$

Part a:

• Argument: $(\sim p \land \sim q) \leftrightarrow (\sim r \lor \sim s)$
• Evaluation:
  • Left side: $\sim p \land \sim q = \text{False} \land \text{True} = \text{False}$
  • Right side: $\sim r \lor \sim s = \text{False} \lor \text{False} = \text{False}$
  • Biconditional: $\text{False} \leftrightarrow \text{False} = \text{True}$
• Answer: True

Part b:

• Argument: $\sim(\sim r) \rightarrow (q \land \sim s)$
• Evaluation:
  • Antecedent: $\sim(\sim r) = r = \text{True}$
  • Consequent: $q \land \sim s = \text{False} \land \text{False} = \text{False}$
  • Implication: $\text{True} \rightarrow \text{False} = \text{False}$
• Answer: False

Part c:

• Argument: $(\sim p \rightarrow \sim r) \land (s \lor \sim(\sim q))$
• Evaluation:
  • First conjunct: $\sim p \rightarrow \sim r = \text{False} \rightarrow \text{False} = \text{True}$
  • Second conjunct: $s \lor \sim(\sim q) = s \lor q = \text{True} \lor \text{False} = \text{True}$
  • Conjunction: $\text{True} \land \text{True} = \text{True}$
• Answer: True

Multiple Choice Assessment Solutions

Question 14: Structural Identification of Cubic Expressions

• Problem Statement: Which of the following products is not the sum or the difference of two perfect cubes?
• Given Options:
  1. $[x + y + z][(x^2 + 2xy + y^2) - z(x + y) + z^2]$
  2. $[a - b + c][(a + c)^2 + ab - bc + b^2]$
  3. $[3x - 5y - 6][6(3x - 5y) + (3x - 5y)^2 + 36]$
  4. $[4x - 2][4 + 8x + 16x^2]$
  5. Cannot be determined by the information given.
• Detailed Structural Analysis:
  • Option (1): Let $A = x + y$ and $B = z$. The expression is $[A + B][A^2 - AB + B^2] = A^3 + B^3 = (x + y)^3 + z^3$. This is the sum of two cubes.
  • Option (2): Let $A = a + c$ and $B = b$. The first bracket is $[A - B]$. For this to be a difference of cubes $A^3 - B^3 = (A - B)(A^2 + AB + B^2)$, the middle term in the second bracket must be $+AB = +b(a + c) = ab + bc$. However, option 2 has $ab - bc = b(a - c)$, which does not match $AB$. Thus, it fails the identity.
  • Option (3): Let $A = 3x - 5y$ and $B = 6$. The expression is $[A - B][AB + A^2 + B^2] = A^3 - B^3 = (3x - 5y)^3 - 216$. This is the difference of two cubes.
  • Option (4): Let $A = 4x$ and $B = 2$. The expression is $[A - B][B^2 + AB + A^2] = A^3 - B^3 = (4x)^3 - 2^3 = 64x^3 - 8$. This is the difference of two cubes.
• Answer: (2)

Question 15: Compound Truth Evaluation

• Problem Statement: If $x$ is true and $y$ is false, then $(\sim x \land y) \lor (x \lor y)$ is:
  • a) true
  • b) false
  • c) neither true or false
  • d) cannot be determined
• Step-by-Step Solution:
  • Evaluate $\sim x$: $\sim(\text{True}) = \text{False}$
  • Evaluate $(\sim x \land y)$: $\text{False} \land \text{False} = \text{False}$
  • Evaluate $(x \lor y)$: $\text{True} \lor \text{False} = \text{True}$
  • Evaluate full disjunction $(\sim x \land y) \lor (x \lor y)$: $\text{False} \lor \text{True} = \text{True}$
• Answer: a) true

Question 16: Conditional Logic Transformations

• Problem Statement: The contrapositive of the statement: "If two lines are parallel, the alternate interior angles are congruent" is:
  • a) If the lines are not parallel then the alternate interior angles are not congruent.
  • b) If the alternate interior angles are not congruent then the lines are not parallel.
  • c) If the alternate interior angles are congruent then the lines are parallel.
  • d) none of these
• Rule Definitions:
  • Original conditional: $P \rightarrow Q$
  • $P$: Two lines are parallel
  • $Q$: Alternate interior angles are congruent
  • Converse: $Q \rightarrow P$ ("If alternate interior angles are congruent then the lines are parallel") — Option (c)
  • Inverse: $\sim P \rightarrow \sim Q$ ("If lines are not parallel then alternate interior angles are not congruent") — Option (a)
  • Contrapositive: $\sim Q \rightarrow \sim P$ ("If alternate interior angles are not congruent then the lines are not parallel") — Option (b)
• Answer: b) If the alternate interior angles are not congruent then the lines are not parallel.