Polynomial Operations Worksheet
Name: ________________________________ Date: ______________ Period: ________
Simplify each expression completely • Write in standard form
1
\((3x^2 + 5x - 4) + (2x^2 - 8x + 9)\)
Answer:
2
\((7y^2 - 2y + 6) - (4y^2 + 5y - 3)\)
Answer:
3
\((4x^3 - 6x^2 + 2x + 7) + (5x^2 - 2x - 10)\)
Answer:
4
\((9a - 4) - (3a - 11)\)
Answer:
5
\((5m^3 + 3m^2 - 7m + 2) + (2m^3 - 8m^2 + 4)\)
Answer:
6
\((6p^3 - p^2 + 4p) - (2p^3 + 5p^2 - 8p + 3)\)
Answer:
7
\((8w^3 - 5w + 12) - (3w^2 + 7w^3 - 9)\)
Answer:
8
\((9k^4 - 2k^3 + 7k - 1) + (3k^3 - 5k^2 - 4k + 8)\)
Answer:
9
\((-4x^3 + 6x^2 - 9x + 5) - (-7x^3 - 2x^2 + 3x - 8)\)
Answer:
10
\((3n^2 - 4n + 7) + (2n^2 + 6n - 5) - (4n^2 - n + 2)\)
Answer:
11
Find the perimeter of a triangle with side lengths \((2x^2 + 3x - 1)\), \((4x^2 - 5)\), and \((x^2 - 2x + 7)\).
Perimeter:
12
A company has revenue \(R(x) = 6x^2 + 14x - 5\) and costs \(C(x) = 2x^2 + 8x + 9\). Find profit \(P(x) = R(x) - C(x)\).
Profit \(P(x)\):
Polynomial Operations Answer Key
Answer Key Polynomial Operations • Page 1 Solutions
Problems 1 – 6 • Standard Form
1
\((3x^2 + 5x - 4) + (2x^2 - 8x + 9)\)
Step-by-Step Solution:
1. Group like terms:
\((3x^2 + 2x^2) + (5x - 8x) + (-4 + 9)\)
2. Combine coefficients:
\(5x^2 + (-3x) + 5\)
Answer: \(5x^2 - 3x + 5\)
2
\((7y^2 - 2y + 6) - (4y^2 + 5y - 3)\)
Step-by-Step Solution:
1. Distribute negative sign:
\(7y^2 - 2y + 6 - 4y^2 - 5y + 3\)
2. Combine like terms:
\((7y^2 - 4y^2) + (-2y - 5y) + (6 + 3)\)
Answer: \(3y^2 - 7y + 9\)
3
\((4x^3 - 6x^2 + 2x + 7) + (5x^2 - 2x - 10)\)
Step-by-Step Solution:
1. Group like terms by power:
\(4x^3 + (-6x^2 + 5x^2) + (2x - 2x) + (7 - 10)\)
2. Note: \(2x - 2x = 0\) (drops out):
\(4x^3 - x^2 + 0x - 3\)
Answer: \(4x^3 - x^2 - 3\)
4
\((9a - 4) - (3a - 11)\)
Step-by-Step Solution:
1. Distribute negative: \(-(-11) = +11\):
\(9a - 4 - 3a + 11\)
2. Combine like terms:
\((9a - 3a) + (-4 + 11) = 6a + 7\)
Answer: \(6a + 7\)
5
\((5m^3 + 3m^2 - 7m + 2) + (2m^3 - 8m^2 + 4)\)
Step-by-Step Solution:
1. Group corresponding powers:
\((5m^3 + 2m^3) + (3m^2 - 8m^2) - 7m + (2 + 4)\)
2. Simplify coefficients:
\(7m^3 - 5m^2 - 7m + 6\)
Answer: \(7m^3 - 5m^2 - 7m + 6\)
6
\((6p^3 - p^2 + 4p) - (2p^3 + 5p^2 - 8p + 3)\)
Step-by-Step Solution:
1. Distribute negative across all 4 terms:
\(6p^3 - p^2 + 4p - 2p^3 - 5p^2 + 8p - 3\)
2. Combine like terms:
\(4p^3 - 6p^2 + 12p - 3\)
Answer: \(4p^3 - 6p^2 + 12p - 3\)
Answer Key Polynomial Operations • Page 2 Solutions
Problems 7 – 12 • Multi-Step & Applications
7
\((8w^3 - 5w + 12) - (3w^2 + 7w^3 - 9)\)
Step-by-Step Solution:
1. Distribute negative:
\(8w^3 - 5w + 12 - 3w^2 - 7w^3 + 9\)
2. Order by descending powers (standard form):
\((8w^3 - 7w^3) - 3w^2 - 5w + (12 + 9)\)
Answer: \(w^3 - 3w^2 - 5w + 21\)
8
\((9k^4 - 2k^3 + 7k - 1) + (3k^3 - 5k^2 - 4k + 8)\)
Step-by-Step Solution:
1. Group corresponding powers:
\(9k^4 + (-2k^3 + 3k^3) - 5k^2 + (7k - 4k) + (-1 + 8)\)
2. Simplify terms:
\(9k^4 + k^3 - 5k^2 + 3k + 7\)
Answer: \(9k^4 + k^3 - 5k^2 + 3k + 7\)
9
\((-4x^3 + 6x^2 - 9x + 5) - (-7x^3 - 2x^2 + 3x - 8)\)
Step-by-Step Solution:
1. Distribute negative across all signs:
\(-4x^3 + 6x^2 - 9x + 5 + 7x^3 + 2x^2 - 3x + 8\)
2. Combine like terms:
\(3x^3 + 8x^2 - 12x + 13\)
Answer: \(3x^3 + 8x^2 - 12x + 13\)
10
\((3n^2 - 4n + 7) + (2n^2 + 6n - 5) - (4n^2 - n + 2)\)
Step-by-Step Solution:
1. Distribute negative to third polynomial:
\(3n^2 - 4n + 7 + 2n^2 + 6n - 5 - 4n^2 + n - 2\)
2. Combine: \((3+2-4)n^2 + (-4+6+1)n + (7-5-2)\)
\(n^2 + 3n + 0\)
Answer: \(n^2 + 3n\)
11
Perimeter of triangle with sides \((2x^2 + 3x - 1)\), \((4x^2 - 5)\), and \((x^2 - 2x + 7)\).
Step-by-Step Solution:
1. Sum of all three side lengths:
\(P = (2x^2 + 3x - 1) + (4x^2 - 5) + (x^2 - 2x + 7)\)
2. Combine like terms:
\((2+4+1)x^2 + (3-2)x + (-1-5+7)\)
Perimeter: \(7x^2 + x + 1\)
12
Revenue \(R(x) = 6x^2 + 14x - 5\) and costs \(C(x) = 2x^2 + 8x + 9\). Find \(P(x) = R(x) - C(x)\).
Step-by-Step Solution:
1. Set up subtraction: \(P(x) = R(x) - C(x)\):
\((6x^2 + 14x - 5) - (2x^2 + 8x + 9)\)
2. Distribute negative & combine:
\(6x^2 - 2x^2 + 14x - 8x - 5 - 9\)
Profit \(P(x)\): \(4x^2 + 6x - 14\)
