Polynomial Operations Answer Key
Teacher Resource • Day 1
Adding & Subtracting Polynomial Functions — Answer Key
Complete Solutions
Bellringer — Day 1 Warm-Up & Pattern Discovery
1. Explain the error, then find the correct simplified expression.
Given Student Work:
3x − 4(x − 2)
3x − 4x − 8 ← Error here!
−1x − 8
Error: The student forgot to distribute the negative to both terms: \(-4(-2) = +8\), not \(-8\).
Correct Work: \(3x - 4x + 8 =\) \(-x + 8\)
2. Look at patterns and fill in missing values.
| x | p(x) | q(x) | r(x) |
|---|
| 0 | 4 | -1 | -4 |
| 1 | 5 | 2 | 10 |
| 2 | 8 | 5 | 40 |
| 3 | 13 | 8 | 104 |
| 4 | 20 | 11 | 220 |
| 5 | 29 | 14 | 406 |
| 6 | 40 | 17 | 680 |
Key: \(p(x) = x^2 + 4\) (+1, +3, +5, +7, +9, +11); \(q(x) = 3x - 1\) (+3 each step); and \(r(x) = p(x) \cdot q(x)\).
Adding and Subtracting Polynomials
Adding Polynomials
Combine like terms. (We are simplifying, not solving.)
Example:
\((2a^3 + 6a^2 - 3) + (5a^3 - 8a^2 + 5)\)
\(= (2a^3 + 5a^3) + (6a^2 - 8a^2) + (-3 + 5)\)
\(= 7a^3 - 2a^2 + 2\)
You Try:
\((4x^3 - 6x^2 + 2x + 7) + (5x^2 - 2x)\)
\(= 4x^3 + (-6x^2 + 5x^2) + (2x - 2x) + 7\)
\(= 4x^3 - x^2 + 7\)
Subtracting Polynomials
Distribute the subtraction, then combine like terms.
Example:
\((2a^3 + 6a^2 - 3) - (5a^3 - 8a^2 + 5)\)
\(= 2a^3 + 6a^2 - 3 - 5a^3 + 8a^2 - 5\)
\(= -3a^3 + 14a^2 - 8\)
You Try:
\((5x - 3) - (2x - 11)\)
\(= 5x - 3 - 2x + 11 = (5x - 2x) + (-3 + 11)\)
\(= 3x + 8\)
Example: Farmer's Field Application Real-World Modeling
A farmer must add the areas of two plots of land to determine the amount of seed to plant. The area of Plot A can be represented by \(3x^2 + 7x - 5\), and Plot B by \(5x^2 - 4x + 11\). Write a polynomial that represents the total area of both plots of land.
\(\text{Total Area} = (3x^2 + 7x - 5) + (5x^2 - 4x + 11)\)
\(= (3x^2 + 5x^2) + (7x - 4x) + (-5 + 11)\)
\(= 8x^2 + 3x + 6\)
Example: Shaded Region Area in terms of \(x\)
2x + 5 6 x + 2 3
Total Area = \(6(2x + 5) = 12x + 30\)
Unshaded = \(3(x + 2) = 3x + 6\)
\(\text{Shaded} = (12x+30) - (3x+6)\)
\(= 12x + 30 - 3x - 6\)
Shaded Area \(= 9x + 24\)
Example: Calculate with Rational Expressions
\(\frac{2x+5}{2} + \frac{9x-4}{3}\)
Common denominator is 6:
\(= \frac{3(2x+5)}{6} + \frac{2(9x-4)}{6}\)
\(= \frac{6x + 15 + 18x - 8}{6}\)
Answer: \(\frac{24x + 7}{6}\) (or \(4x + \frac{7}{6}\))
Algebra 1 / Math 2 • Unit: Polynomial Functions Page 1 of 2
Teacher Resource • Day 2
Function Notation & Combined Functions — Answer Key
Complete Solutions
Bellringer — Day 2 Linear Function Evaluation
1. Use the function rule to complete the table below: \(u(x) = 2x - 3\)
| x | -2 | -1 | 0 | 5 |
|---|
| u(x) = 2x − 3 | -7 | -5 | -3 | 7 |
Steps: \(u(-2) = 2(-2)-3 = -7\); \(u(-1) = 2(-1)-3 = -5\); \(u(0) = 2(0)-3 = -3\); \(u(5) = 2(5)-3 = 7\).
Adding and Subtracting Functions Notation: \((f \pm g)(x) = f(x) \pm g(x)\)
Adding: \(f(x) + g(x) = h(x) \iff (f+g)(x) = h(x)\)
Example:
Write function rule for \(w(x)\), if \(u(x)=2x-3\) and \(v(x)=3x+1\) and \(w(x)=u(x)+v(x)\).
\(w(x) = (2x - 3) + (3x + 1)\)
\(w(x) = (2x + 3x) + (-3 + 1)\)
\(w(x) = 5x - 2\)
Subtracting: \(f(x) - g(x) = h(x) \iff (f-g)(x) = h(x)\)
Example:
Write function rule for \(w(x)\), if \(u(x)=2x-3\) and \(v(x)=3x+1\) and \(w(x)=u(x)-v(x)\).
\(w(x) = (2x - 3) - (3x + 1)\)
\(w(x) = 2x - 3 - 3x - 1\)
\(w(x) = -x - 4\)
You Try (Order Matters):
Write rule for \(w(x)\) if \(w(x) = v(x) - u(x)\):
\(w(x) = (3x + 1) - (2x - 3) = 3x + 1 - 2x + 3\)
\(w(x) = x + 4\)
Tables & Graphical Representation Determine \(f(x)\), \(g(x)\), and \(h(x)\)
| x | -2 | -1 | 0 | 1 | 2 |
|---|
| f(x) = 3x + 1 | -5 | -2 | 1 | 4 | 7 |
| g(x) = 2x | -4 | -2 | 0 | 2 | 4 |
| h(x) = f(x) + g(x) | -9 | -4 | 1 | 6 | 11 |
What is the function for \(h(x)\)?
\(h(x) = f(x) + g(x) = (3x + 1) + 2x\)
\(h(x) = 5x + 1\)
Graph & Predict \(h(x)\): Teacher Explanation
Key Concept / Prediction: Adding two linear functions results in another linear function whose slope is the sum of the individual slopes (\(m = 3 + 2 = 5\)) and whose y-intercept is the sum of their intercepts (\(b = 1 + 0 = 1\)).
- f(x) = 3x + 1: Passes through \((0, 1)\) with slope 3.
- g(x) = 2x: Passes through origin \((0, 0)\) with slope 2.
- h(x) = 5x + 1: Shares y-intercept \((0, 1)\) with \(f(x)\), but rises significantly steeper (slope 5). For every \(x\), the height of \(h(x)\) equals the vertical sum of \(f(x)\) and \(g(x)\).
f(x) g(x) h(x)
Algebra 1 / Math 2 • Unit: Polynomial Functions Page 2 of 2
