Solved Exercises on Adding & Subtracting Polynomials in Algebra 2

Master adding and subtracting polynomials: combining like terms, distributing negative signs, and simplifying polynomial expressions through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Mixed operations with polynomials
Exercise 1
Simplify: \( (3x^2 + 2x - 5) + (x^2 - 3x + 4) - (2x^2 + x - 1) \)
Definition:

Polynomial Operations: When combining multiple operations, perform addition and subtraction by combining like terms after properly distributing signs.

Method for mixed polynomial operations:
  1. Handle subtraction by distributing negative signs to all terms in parentheses
  2. Remove all parentheses
  3. Group like terms together
  4. Add or subtract coefficients of like terms
  5. Write result in standard form
Step 1: Handle the subtraction by distributing the negative sign

\( (3x^2 + 2x - 5) + (x^2 - 3x + 4) - (2x^2 + x - 1) \)

\( = (3x^2 + 2x - 5) + (x^2 - 3x + 4) + (-(2x^2 + x - 1)) \)

\( = (3x^2 + 2x - 5) + (x^2 - 3x + 4) + (-2x^2 - x + 1) \)

Step 2: Remove parentheses

\( 3x^2 + 2x - 5 + x^2 - 3x + 4 - 2x^2 - x + 1 \)

Step 3: Group like terms together

\( (3x^2 + x^2 - 2x^2) + (2x - 3x - x) + (-5 + 4 + 1) \)

Step 4: Add coefficients of like terms

\( (3 + 1 - 2)x^2 + (2 - 3 - 1)x + (-5 + 4 + 1) \)

\( 2x^2 + (-2)x + 0 \)

\( 2x^2 - 2x \)

Step 5: Write in standard form

\( 2x^2 - 2x \)

\( (3x^2 + 2x - 5) + (x^2 - 3x + 4) - (2x^2 + x - 1) = 2x^2 - 2x \)
Final answer:

\( 2x^2 - 2x \)

Applied rules:

• Sign Distribution: Multiply each term in the subtracted polynomial by -1

• Like Terms: Combine terms with identical variable parts

• Standard Form: Arrange terms in descending order of degree

2 Higher degree polynomials
Exercise 2
Simplify: \( (4x^3 - 2x^2 + x - 3) - (x^3 + 3x^2 - 2x + 1) + (2x^2 - x + 5) \)
Definition:

Higher Degree Polynomial Operations: The same principles apply to polynomials of any degree - distribute signs, combine like terms, and arrange in standard form.

Step 1: Handle the subtraction by distributing the negative sign

\( (4x^3 - 2x^2 + x - 3) + (-(x^3 + 3x^2 - 2x + 1)) + (2x^2 - x + 5) \)

\( (4x^3 - 2x^2 + x - 3) + (-x^3 - 3x^2 + 2x - 1) + (2x^2 - x + 5) \)

Step 2: Remove parentheses

\( 4x^3 - 2x^2 + x - 3 - x^3 - 3x^2 + 2x - 1 + 2x^2 - x + 5 \)

Step 3: Group like terms by degree

\( (4x^3 - x^3) + (-2x^2 - 3x^2 + 2x^2) + (x + 2x - x) + (-3 - 1 + 5) \)

Step 4: Add coefficients of like terms

\( (4 - 1)x^3 + (-2 - 3 + 2)x^2 + (1 + 2 - 1)x + (-3 - 1 + 5) \)

\( 3x^3 + (-3)x^2 + 2x + 1 \)

\( 3x^3 - 3x^2 + 2x + 1 \)

Step 5: Verify by checking each degree

3rd degree: \( 4x^3 - x^3 = 3x^3 \) ✓

2nd degree: \( -2x^2 - 3x^2 + 2x^2 = -3x^2 \) ✓

1st degree: \( x + 2x - x = 2x \) ✓

Constant: \( -3 - 1 + 5 = 1 \) ✓

\( (4x^3 - 2x^2 + x - 3) - (x^3 + 3x^2 - 2x + 1) + (2x^2 - x + 5) = 3x^3 - 3x^2 + 2x + 1 \)
Final answer:

\( 3x^3 - 3x^2 + 2x + 1 \)

Applied rules:

• Complete Distribution: Every term in the subtracted polynomial gets multiplied by -1

• Like Terms: Group by identical variable parts

• Standard Form: Terms arranged in descending order of degree

3 Operations with non-standard form
Exercise 3
Simplify: \( (2x - 3x^3 + x^2 + 4) - (x^2 - 4x^3 + 2x - 1) + (3x^3 - x + 5) \)
Definition:

Non-Standard Form Polynomials: When polynomials are not arranged in descending order, it's helpful to rearrange them first before performing operations.

Step 1: Rearrange each polynomial in standard form

First: \( -3x^3 + x^2 + 2x + 4 \)

Second: \( -4x^3 + x^2 + 2x - 1 \)

Third: \( 3x^3 - x + 5 \)

Step 2: Rewrite with operations

\( (-3x^3 + x^2 + 2x + 4) - (-4x^3 + x^2 + 2x - 1) + (3x^3 - x + 5) \)

Step 3: Distribute the negative sign to the second polynomial

\( (-3x^3 + x^2 + 2x + 4) + (4x^3 - x^2 - 2x + 1) + (3x^3 - x + 5) \)

Step 4: Remove parentheses and group like terms

\( -3x^3 + x^2 + 2x + 4 + 4x^3 - x^2 - 2x + 1 + 3x^3 - x + 5 \)

\( (-3x^3 + 4x^3 + 3x^3) + (x^2 - x^2) + (2x - 2x - x) + (4 + 1 + 5) \)

Step 5: Add coefficients of like terms

\( (-3 + 4 + 3)x^3 + (1 - 1)x^2 + (2 - 2 - 1)x + (4 + 1 + 5) \)

\( 4x^3 + 0x^2 + (-1)x + 10 \)

\( 4x^3 - x + 10 \)

\( (2x - 3x^3 + x^2 + 4) - (x^2 - 4x^3 + 2x - 1) + (3x^3 - x + 5) = 4x^3 - x + 10 \)
Final answer:

\( 4x^3 - x + 10 \)

Applied rules:

• Standard Form: Arrange terms in descending order of degree

• Sign Distribution: Multiply each term in the subtracted polynomial by -1

• Like Terms: Combine terms with identical variable parts

Rules and methods, laws,...
\( A(x) ± B(x) ± C(x) = (a_n ± b_n ± c_n)x^n + ... + (a_1 ± b_1 ± c_1)x + (a_0 ± b_0 ± c_0) \)
General Polynomial Operation Formula
Addition
A + B
Combine like terms
Subtraction
A - B = A + (-B)
Distribute negative
Like Terms
Same variable parts
Add coefficients
Standard Form
Descending order
Degrees from high to low
Key definitions:

Polynomial Addition: Combining polynomials by adding coefficients of like terms.

Polynomial Subtraction: Finding the difference by adding the first polynomial to the negative of the second polynomial.

Like Terms: Terms with identical variable parts (same variables raised to the same powers).

Standard Form: A polynomial written with terms in descending order of degree.

Distribution: Multiplying each term inside parentheses by the factor outside.

Leading Term: The term with the highest degree in a polynomial written in standard form.

Coefficient: The numerical factor of a term.

Steps for adding and subtracting polynomials:
  1. Identify operations: Note which polynomials are being added and which are being subtracted
  2. Distribute signs: For subtraction, multiply each term in the polynomial by -1
  3. Remove parentheses: After distributing signs
  4. Group like terms: Arrange terms with the same degree together
  5. Combine like terms: Add or subtract coefficients of like terms
  6. Write in standard form: Arrange terms from highest to lowest degree
Properties: Addition is commutative and associative; subtraction is neither; the result is always a polynomial; degree of result is at most the highest degree of original polynomials.
Applications: Combining functions in mathematics, physics, engineering; modeling combined effects; simplifying complex expressions; solving polynomial equations.
Tip 1: Always distribute the negative sign to ALL terms in the polynomial being subtracted.
Tip 2: Arrange polynomials in standard form before operating to avoid missing like terms.
Tip 3: Use the vertical method to align like terms, especially for longer polynomials.
Important rules to know:

• Subtraction Conversion: A(x) - B(x) = A(x) + (-B(x))

• Complete Distribution: Every term in the subtracted polynomial must be multiplied by -1

• Like Terms Only: Only combine terms with identical variable parts

• Standard Form: Write final answer with terms in descending order of degree

• Commutativity: Addition is commutative, subtraction is not

Solution: Exercises 4 to 5
4 Operations with multiple variables
Exercise 4
Simplify: \( (3x^2y - 2xy^2 + x + 4) + (x^2y + 5xy^2 - 2x + 3) - (2x^2y - xy^2 + x - 1) \)
Definition:

Polynomials with Multiple Variables: Like terms have identical variable parts with the same exponents for each variable.

Step 1: Handle the subtraction by distributing the negative sign

\( (3x^2y - 2xy^2 + x + 4) + (x^2y + 5xy^2 - 2x + 3) + (-(2x^2y - xy^2 + x - 1)) \)

\( (3x^2y - 2xy^2 + x + 4) + (x^2y + 5xy^2 - 2x + 3) + (-2x^2y + xy^2 - x + 1) \)

Step 2: Remove parentheses

\( 3x^2y - 2xy^2 + x + 4 + x^2y + 5xy^2 - 2x + 3 - 2x^2y + xy^2 - x + 1 \)

Step 3: Group like terms together

\( (3x^2y + x^2y - 2x^2y) + (-2xy^2 + 5xy^2 + xy^2) + (x - 2x - x) + (4 + 3 + 1) \)

Step 4: Add coefficients of like terms

\( (3 + 1 - 2)x^2y + (-2 + 5 + 1)xy^2 + (1 - 2 - 1)x + (4 + 3 + 1) \)

\( 2x^2y + 4xy^2 + (-2)x + 8 \)

\( 2x^2y + 4xy^2 - 2x + 8 \)

Step 5: Verify by checking each term type

\( x^2y \) terms: \( 3x^2y + x^2y - 2x^2y = 2x^2y \) ✓

\( xy^2 \) terms: \( -2xy^2 + 5xy^2 + xy^2 = 4xy^2 \) ✓

\( x \) terms: \( x - 2x - x = -2x \) ✓

Constants: \( 4 + 3 + 1 = 8 \) ✓

\( (3x^2y - 2xy^2 + x + 4) + (x^2y + 5xy^2 - 2x + 3) - (2x^2y - xy^2 + x - 1) = 2x^2y + 4xy^2 - 2x + 8 \)
Final answer:

\( 2x^2y + 4xy^2 - 2x + 8 \)

Applied rules:

• Multiple Variable Like Terms: Same variables with same exponents

• Complete Distribution: Every term in the subtracted polynomial gets multiplied by -1

• Combine Coefficients: Add only the numerical parts

5 Real-world application
Exercise 5
The revenue of a company is modeled by \( R(x) = 3x^2 + 5x + 100 \) thousand dollars, the cost of production by \( C(x) = x^2 + 2x + 50 \) thousand dollars, and taxes by \( T(x) = 0.5x^2 + x + 15 \) thousand dollars, where x is the number of items produced in hundreds. Find the net profit after taxes: \( P(x) = R(x) - C(x) - T(x) \).
Definition:

Net Profit After Taxes: The actual profit after accounting for both costs and tax obligations, calculated as revenue minus costs minus taxes.

Step 1: Set up the expression for net profit

\( P(x) = R(x) - C(x) - T(x) \)

\( P(x) = (3x^2 + 5x + 100) - (x^2 + 2x + 50) - (0.5x^2 + x + 15) \)

Step 2: Convert to addition of negatives

\( P(x) = (3x^2 + 5x + 100) + (-(x^2 + 2x + 50)) + (-(0.5x^2 + x + 15)) \)

Step 3: Distribute the negative signs

\( -(x^2 + 2x + 50) = -x^2 - 2x - 50 \)

\( -(0.5x^2 + x + 15) = -0.5x^2 - x - 15 \)

Step 4: Write the complete expression

\( P(x) = (3x^2 + 5x + 100) + (-x^2 - 2x - 50) + (-0.5x^2 - x - 15) \)

Step 5: Remove parentheses and group like terms

\( 3x^2 + 5x + 100 - x^2 - 2x - 50 - 0.5x^2 - x - 15 \)

\( (3x^2 - x^2 - 0.5x^2) + (5x - 2x - x) + (100 - 50 - 15) \)

Step 6: Add coefficients of like terms

\( (3 - 1 - 0.5)x^2 + (5 - 2 - 1)x + (100 - 50 - 15) \)

\( 1.5x^2 + 2x + 35 \)

Step 7: Interpret the result

\( P(x) = 1.5x^2 + 2x + 35 \) represents the net profit in thousands of dollars

When x hundred items are produced, the company makes \( 1.5x^2 + 2x + 35 \) thousand dollars in net profit.

Net profit: \( P(x) = 1.5x^2 + 2x + 35 \) thousand dollars
Final answer:

The net profit after taxes is modeled by \( P(x) = 1.5x^2 + 2x + 35 \) thousand dollars.

Applied rules:

• Business Application: Net Profit = Revenue - Costs - Taxes

• Polynomial Operations: Properly distribute signs and combine like terms

• Like Terms: Combine terms with the same degree

Comprehensive Summary: Adding & Subtracting Polynomials
\( A(x) ± B(x) ± C(x) = (a_n ± b_n ± c_n)x^n + ... + (a_1 ± b_1 ± c_1)x + (a_0 ± b_0 ± c_0) \)
General Polynomial Operation Formula
Key definitions:

Polynomial Addition: The operation of combining two or more polynomials by adding their like terms. The result is always another polynomial.

Polynomial Subtraction: The operation of finding the difference between polynomials by adding the first polynomial to the negative of the second polynomial.

Like Terms: Terms that have identical variable parts (same variables raised to the same powers). Only like terms can be combined.

Standard Form: A polynomial written with terms arranged in descending order of degree (highest degree first).

Distribution: The process of multiplying each term inside parentheses by the factor outside, especially important when subtracting polynomials.

Coefficient: The numerical factor of a term. When adding or subtracting, we add or subtract coefficients while keeping the variable part unchanged.

Leading Term: The term with the highest degree in a polynomial written in standard form.

Systematic approach for polynomial operations:
  1. Identify operations: Determine which polynomials are being added and which are being subtracted
  2. Convert subtractions: Change A - B to A + (-B)
  3. Distribute signs: Multiply each term in subtracted polynomials by -1
  4. Remove parentheses: After distributing signs
  5. Organize: Write each polynomial in standard form
  6. Group like terms: Arrange terms with the same degree together
  7. Combine like terms: Add or subtract coefficients of like terms
  8. Arrange in standard form: Order terms from highest to lowest degree
  9. Verify: Check that all terms have been properly accounted for
Tip 1: Always distribute the negative sign to ALL terms in the polynomial being subtracted, not just the first one.
Tip 2: Use the vertical method to align like terms, especially for longer polynomials.
Tip 3: Remember that subtracting a negative term is equivalent to adding a positive term.
Tip 4: For polynomials with multiple variables, like terms must have identical variable parts.
Common Mistakes: Forgetting to distribute the negative to all terms, incorrectly changing signs, adding instead of subtracting like terms, not arranging in standard form, arithmetic errors when combining coefficients.
Exam Preparation: Practice with polynomials of different degrees, master the concept of like terms, become proficient with integer arithmetic, understand that subtraction is not commutative.
Essential rules and properties:

• Subtraction Conversion: A(x) - B(x) = A(x) + (-B(x))

• Complete Distribution: Every term in the subtracted polynomial must be multiplied by -1

• Like Terms Only: Combine terms with identical variable parts

• Coefficient Operations: Add or subtract coefficients while preserving variable parts

• Closure Property: The result of polynomial addition/subtraction is always a polynomial

• Commutativity: Addition is commutative (A + B = B + A), subtraction is not (A - B ≠ B - A)

• Associativity: Addition is associative (A + (B + C) = (A + B) + C)

Polynomial Operations Concepts and Techniques

Polynomial Operations Process

Distribute Signs

Multiply by -1

→

Group Like Terms

Same variable parts

→

Combine Coefficients

Add or subtract

Key Principles:

  • Every term in subtracted polynomials gets multiplied by -1
  • Only combine terms with identical variable parts
  • Add or subtract coefficients, preserve variables
  • Result is always a polynomial
  • Arranged in descending degree order

Key insight: Polynomial operations build upon the fundamental concept of combining like terms, with proper sign handling being crucial for accuracy.

Questions & Answers

Question: I get confused about the order of operations when there are multiple additions and subtractions. Should I go left to right?

Answer: Yes, you can work left to right, but there's a more strategic approach:

Left-to-Right Method:

  • Work through the expression from left to right
  • Handle each operation as you encounter it
  • For example: A + B - C = (A + B) - C

Grouping Method (Recommended):

  • Think of it as: first polynomial + (second polynomial) + (third polynomial with negative distributed)
  • For example: A + B - C = A + B + (-C)
  • This allows you to combine all like terms at once

Example: \( (2x^2 + 3x - 1) - (x^2 - 2x + 4) + (3x^2 + x - 2) \)

  • Convert to: \( (2x^2 + 3x - 1) + (-x^2 + 2x - 4) + (3x^2 + x - 2) \)
  • Combine all at once: \( (2x^2 - x^2 + 3x^2) + (3x + 2x + x) + (-1 - 4 - 2) \)
  • Result: \( 4x^2 + 6x - 7 \)

The grouping method is more efficient as it reduces the number of intermediate steps.

Question: What happens when I subtract two polynomials and all terms cancel out? Is the answer 0?

Answer: Yes, if all terms cancel out, the result is the zero polynomial:

Example: \( (3x^2 + 2x - 5) - (3x^2 + 2x - 5) \)

  • Distribute: \( (3x^2 + 2x - 5) + (-3x^2 - 2x + 5) \)
  • Combine: \( (3x^2 - 3x^2) + (2x - 2x) + (-5 + 5) \)
  • Result: \( 0x^2 + 0x + 0 = 0 \)

When this happens:

  • The two polynomials were identical
  • The result is the zero polynomial (denoted as 0)
  • The degree of the zero polynomial is undefined

Another example: \( (x^3 - 2x^2 + x - 1) - (x^3 - 2x^2 + x - 1) = 0 \)

This is perfectly valid and correct. When two identical polynomials are subtracted, the result is always zero.

Question: Can I add or subtract polynomials of different degrees? Will the result always be the highest degree?

Answer: Yes, you can add or subtract polynomials of different degrees, but the degree of the result depends on whether the leading terms cancel:

Possible Outcomes:

  • Degree preserved: If leading terms don't cancel, the result has the highest degree of the original polynomials
  • Degree reduced: If leading terms cancel, the result has a lower degree
  • Zero result: If all terms cancel, the result is 0 (undefined degree)

Examples:

  • Addition: \( (x^3 + 2x) + (x^2 + 3) = x^3 + x^2 + 2x + 3 \) (degree 3 preserved)
  • Subtraction: \( (x^3 + 2x) - (x^2 + 3) = x^3 - x^2 + 2x - 3 \) (degree 3 preserved)
  • Subtraction: \( (x^3 + 2x^2) - (x^3 + x^2) = x^2 \) (degree reduced from 3 to 2)

Key Point: The degree of the result is at most the maximum of the degrees of the polynomials being operated on.

This is different from multiplication, where degrees are added.

Question: Is there a way to check my answer after adding or subtracting polynomials?

Answer: Yes, here are several methods to check your polynomial operations:

Method 1: Substitution

  • Choose a value for x (like x = 1 or x = 2)
  • Substitute into the original polynomials and compute
  • Substitute into your result and compare

Example: Check \( (x^2 + 2x + 1) + (x^2 - x + 3) = 2x^2 + x + 4 \)

  • Let x = 1: Original = (1 + 2 + 1) + (1 - 1 + 3) = 4 + 3 = 7
  • Result: 2(1)² + 1 + 4 = 2 + 1 + 4 = 7 ✓

Method 2: Reverse Operation

  • Take your result and perform the inverse operation
  • You should get back to the original polynomial

Method 3: Term-by-Term Verification

  • Check that each degree term was computed correctly
  • Verify sign distribution for subtraction
  • Confirm like terms were properly grouped

The substitution method is particularly effective for catching arithmetic errors.

Question: How do I handle fractions or decimals as coefficients when adding/subtracting polynomials?

Answer: Fractional or decimal coefficients are handled the same way as integer coefficients:

With Fractions:

  • Find common denominators when adding/subtracting fractions
  • Apply the same polynomial operation rules

Example: \( \left(\frac{1}{2}x^2 + \frac{3}{4}x + 1\right) + \left(\frac{1}{3}x^2 - \frac{1}{4}x + 2\right) \)

  • \( x^2 \) terms: \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \)
  • \( x \) terms: \( \frac{3}{4} + \left(-\frac{1}{4}\right) = \frac{2}{4} = \frac{1}{2} \)
  • Constants: \( 1 + 2 = 3 \)
  • Result: \( \frac{5}{6}x^2 + \frac{1}{2}x + 3 \)

With Decimals:

  • Align decimal points when adding/subtracting
  • Round appropriately if needed for the context

Key Point: The variable parts remain unchanged - only the coefficients are affected by the arithmetic. The same rules for like terms and standard form apply regardless of coefficient type.