Polynomial Showdown Practice Worksheet
AR.4A Algebra I • Operations on Polynomials
Question 3 Practice (Subtract Polynomials)
Name:
Date:
Period:
Standard Strategy: When subtracting \(j(x) = h(x) - k(x)\), write both polynomials in parentheses: \((h(x)) - (k(x))\). Remember to distribute the negative sign to every term inside \(k(x)\) before combining like terms.
3C. Independent Practice
If \(h(x) = 8x - 72\) and \(k(x) = 5x^2 - 3x + 18\), which represents \(j(x) = h(x) - k(x)\)?
A \(j(x) = -5x^2 + 11x - 90\)
B \(j(x) = -5x^2 + 5x - 54\)
C \(j(x) = 5x^2 - 11x + 90\)
D \(j(x) = -5x^2 + 11x - 54\)
Show Work (Distribute & Combine):
3D. Independent Practice
If \(h(x) = 3x - 36\) and \(k(x) = 4x^2 - 8x + 24\), which represents \(j(x) = h(x) - k(x)\)?
A \(j(x) = 4x^2 - 11x + 60\)
B \(j(x) = -4x^2 + 11x - 60\)
C \(j(x) = -4x^2 - 5x - 12\)
D \(j(x) = -4x^2 + 11x - 12\)
Show Work (Distribute & Combine):
Trap Detector: Error Analysis
In Question 3A, a student evaluated \(j(x) = (4x - 60) - (3x^2 - 5x + 40)\) and arrived at \(-3x^2 - x - 20\). Explain the two specific sign errors this student made when distributing the negative sign.
Error 1 (Linear term):
Error 2 (Constant term):
Standard AR.4A Focus • Linear Binomial Minus Quadratic Trinomial Target: Avoid Negative Sign Distribution Traps
Polynomial Showdown Answer Key
TEACHER KEY AR.4A • Polynomial Operations
Solutions & Distractor Rationales
Core Assessment Objective: State standard AR.4A tests whether students remember that subtracting a polynomial requires distributing \(-1\) across all terms of the subtrahend, and reordering the result into standard form \(ax^2 + bx + c\).
Problem 3C Solution Correct: Choice A
\(j(x) = (8x - 72) - (5x^2 - 3x + 18)\)
\(\phantom{j(x)} = 8x - 72 - 5x^2 + 3x - 18\)
\(\phantom{j(x)} = -5x^2 + (8x + 3x) + (-72 - 18)\)
\(\phantom{j(x)} = -5x^2 + 11x - 90\)
Distractor Explanations:
A (Correct): Correctly distributes \(-1\) to \(-3x\) and \(+18\).
B: Failed to distribute: calculated \(8x - 3x = 5x\) and \(-72 + 18 = -54\).
C: Subtracted in reverse order \(k(x) - h(x)\).
D: Distributed to \(x\) term but forgot constant (\(-72 + 18 = -54\)).
Problem 3D Solution Correct: Choice B
\(j(x) = (3x - 36) - (4x^2 - 8x + 24)\)
\(\phantom{j(x)} = 3x - 36 - 4x^2 + 8x - 24\)
\(\phantom{j(x)} = -4x^2 + (3x + 8x) + (-36 - 24)\)
\(\phantom{j(x)} = -4x^2 + 11x - 60\)
Distractor Explanations:
A: Subtracted in reverse order \(k(x) - h(x)\).
B (Correct): Correctly distributes \(-1\) to \(-8x\) and \(+24\).
C: Failed to distribute: calculated \(3x - 8x = -5x\) and \(-36 + 24 = -12\).
D: Distributed to \(x\) term but forgot constant (\(-36 + 24 = -12\)).
Trap Detector Key: Question 3A Error Analysis Prompt Error: \(-3x^2 - x - 20\)
Error 1 (Linear term): The student computed \(4x - 5x = -x\) because they subtracted directly without distributing the negative sign to \(-5x\). It should be \(4x - (-5x) = 4x + 5x = \mathbf{9x}\).
Error 2 (Constant term): The student computed \(-60 + 40 = -20\) because they did not distribute the negative to \(+40\). It should be \(-60 - (+40) = \mathbf{-100}\).
Correct 3A Solution: \(j(x) = (4x - 60) - (3x^2 - 5x + 40) = \mathbf{-3x^2 + 9x - 100}\).
Classroom Coaching Notes for AR.4A
1. Scaffold with Parentheses: Train students to immediately rewrite the subtraction with grouping symbols: \(\mathbf{[h(x)] - [k(x)]}\). Many students write \(4x - 60 - 3x^2 - 5x + 40\), which drops the crucial grouping.
2. Watch the Order of Operands: Notice that \(h(x)\) is linear and \(k(x)\) is quadratic. Students instinctively put the quadratic first, inadvertently computing \(k(x) - h(x)\) and getting the opposite sign on all terms.
Teacher Reference • TEKS AR.4A / Standard Polynomial Operations Page 1 of 1
