NC Math 3 Released Items Study Guide: Polynomials and Rational Expressions

NC Math 3 Released Items: Exam Guidelines and Gridded Response Procedure

• Gridded Response Item Instructions:     * Questions 1 and 2 are categorized as gridded response items.     * Answering Procedure:         * Write answers directly in the boxes provided on the answer sheet.         * Restriction: Write only one number or symbol per box.         * Column Matching: Fill in the circle in each column that corresponds to the number or symbol printed in the box above it.         * Limitation: Fill in only one circle per column.

Mathematical Analysis: Polynomial Division and Remainder Theorem

• Objective: Determine the remainder of a cubic polynomial division.
• Problem Statement: Find the remainder when $x^3 - 1$ is divided by $(x + 2)$.
• Theoretical Framework (Remainder Theorem):     * The Remainder Theorem states that if a polynomial $f(x)$ is divided by a linear factor of the form $(x - c)$, the resulting remainder is equal to $f(c)$.
• Application:     * Define the function: $f(x) = x^3 - 1$     * Identify the divisor: $(x + 2)$, which implies $c = -2$.     * Substitute $c$ into the function:         * $f(-2) = (-2)^3 - 1$         * $f(-2) = -8 - 1$         * $f(-2) = -9$
• Conclusion: The remainder is $-9$.

Complex Functions: Determining Real and Imaginary Zeros

• Objective: Identify the number of real zeros in a fifth-degree polynomial.
• Problem Statement: Given the function $f(x) = 2x^5 - 13x^4 + 22x^3 - 187x^2 - 160x + 336$, find the total number of real zeros.
• Given Factors:     * $(x - 7)$     * $(x + 4i)$
• Algebraic Properties:     * Fundamental Theorem of Algebra: A polynomial of degree $n$ has exactly $n$ complex zeros (including multiplicities).     * Complex Conjugate Root Theorem: If a complex number $a + bi$ is a root of a polynomial with real coefficients, then its conjugate $a - bi$ is also a root.
• Root Analysis:     * The degree is $5$, indicating a total of 5 zeros.     * Factor $(x - 7)$ yields a real zero: $x = 7$.     * Factor $(x + 4i)$ yields a complex zero: $x = -4i$.     * By the Complex Conjugate Root Theorem, because $-4i$ is a root, $+4i$ (from the factor $(x - 4i)$) must also be a root.     * Current zero count: $1$ real ($7$), $2$ complex ($4i, -4i$).
• Polynomial Reduction:     * The product of the complex factors is $(x + 4i)(x - 4i) = x^2 + 16$.     * The product of known factors is $(x - 7)(x^2 + 16) = x^3 - 7x^2 + 16x - 112$.     * Dividing the original polynomial $f(x)$ by $(x^3 - 7x^2 + 16x - 112)$ results in the quotient $2x^2 + x - 3$.
• Finding Remaining Zeros:     * Solve $2x^2 + x - 3 = 0$ using the quadratic formula: x = \frac{-b \text{ } pm \text{ } \text{\sqrt{b^2 - 4ac}}}{2a}.     * Discriminant calculation: \text{\Delta} = 1^2 - 4(2)(-3) = 1 + 24 = 25.     * Since the discriminant is positive and a perfect square, there are two distinct real rational zeros.     * Calculation: x = \frac{-1 \text{ } pm \text{ } 5}{4}, resulting in $x = 1$ and $x = -1.5$.
• Final Tally:     * Real Zeros: $7, 1, -1.5$ (Total: 3).     * Complex Zeros: $4i, -4i$ (Total: 2).
• Conclusion: The total number of real zeros is $3$.

Solving Cubic Polynomials via Synthetic Division and Quadratic Formula

• Objective: Identify all zeros of a cubic function given one known zero.
• Problem Statement: The graph of $m(x) = x^3 + 3x^2 - 2x - 4$ has a zero at $x = 1$. Identify the remaining zeros.
• Note on Transcript Consistency: While the transcript states a zero at $1$, the function evaluation $m(1) = 1 + 3 - 2 - 4 = -2$ suggests the transcript contains a common typographical error where the zero should be at $-1$. Following the standard NC Math 3 curriculum solution for this specific problem (Option C):
• Step-by-Step Division (assuming zero at $-1$):     * Perform (x^3 + 3x^2 - 2x - 4) \text{ } \text{\div} \text{ } (x + 1).     * Quotient: $x^2 + 2x - 4$.
• Solving the Quadratic Quotient:     * Set $x^2 + 2x - 4 = 0$.     * Use the quadratic formula: x = \frac{-2 \text{ } pm \text{ } \text{\sqrt{2^2 - 4(1)(-4)}}}{2(1)}.     * x = \frac{-2 \text{ } pm \text{ } \text{\sqrt{4 + 16}}}{2}.     * x = \frac{-2 \text{ } pm \text{ } \text{\sqrt{20}}}{2}.     * Simplify the radical: \text{\sqrt{20}} = 2\text{\sqrt{5}}.     * x = \frac{-2 \text{ } pm \text{ } 2\text{\sqrt{5}}}{2} = -1 \text{ } pm \text{ } \text{\sqrt{5}}.
• Options Provided:     * A: $-2$ and $2$     * B: $-1$ and $4$     * C: -1 \text{ } pm \text{ } \text{\sqrt{5}}     * D: 1 \text{ } pm \text{ } 2\text{\sqrt{3}}
• Conclusion: The correct other zeros are -1 \text{ } pm \text{ } \text{\sqrt{5}} (Option C).

Rational Expressions: Simplification and Polynomial Long Division

• Objective: Simplify a rational expression by performing division.
• Problem Statement: Which expression is equivalent to (x^2 - 2x - 37) \text{ } \text{\div} \text{ } (x^2 - 3x - 40), where both numerator and denominator are quadratic trinomials?
• Execution (Polynomial Long Division):     * Divide the lead coefficient: x^2 \text{ } \text{\div} \text{ } x^2 = 1.     * Multiply the entire divisor by $1$: $1(x^2 - 3x - 40) = x^2 - 3x - 40$.     * Subtract this from the original numerator:         * $(x^2 - 2x - 37) - (x^2 - 3x - 40)$         * $x^2 - x^2 - 2x - (-3x) - 37 - (-40)$         * $x + 3$
• Formulating the Result:     * The result is presented as $\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$.     * Expression: $1 + \frac{x + 3}{x^2 - 3x - 40}$.
• Options Provided:     * A: $1 + \frac{x + 3}{x^2 - 3x - 40}$     * B: $1 - \frac{x + 3}{x^2 - 3x - 40}$     * C: $1 + \frac{2x - 37}{x^2 - 3x - 40}$     * D: $1 - \frac{2x - 37}{x^2 - 3x - 40}$
• Conclusion: The equivalent expression is Option A.