Solved Exercises on Subtracting Polynomials (Intro) in Algebra 2

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

Solution: Exercises 1 to 3
1 Basic polynomial subtraction
Exercise 1
Subtract the polynomials: \( (5x^2 + 3x - 2) - (2x^2 - x + 4) \)
Definition:

Polynomial Subtraction: Subtracting polynomials is equivalent to adding the first polynomial to the negative of the second polynomial: \( A(x) - B(x) = A(x) + (-B(x)) \).

Method for subtracting polynomials:
  1. Change subtraction to addition of the negative: \( A - B = A + (-B) \)
  2. Distribute the negative sign to all terms in the second polynomial
  3. Combine like terms by adding coefficients
  4. Write the result in standard form
Step 1: Convert subtraction to addition of the negative

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

Step 2: Distribute the negative sign to the second polynomial

\( -(2x^2 - x + 4) = -2x^2 + x - 4 \)

Step 3: Rewrite the expression

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

Step 4: Remove parentheses and group like terms

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

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

Step 5: Add coefficients of like terms

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

\( 3x^2 + 4x - 6 \)

\( (5x^2 + 3x - 2) - (2x^2 - x + 4) = 3x^2 + 4x - 6 \)
Final answer:

\( 3x^2 + 4x - 6 \)

Applied rules:

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

• Like Terms: Combine terms with identical variable parts

• Standard Form: Arrange terms in descending order of degree

2 Subtraction with higher degree
Exercise 2
Subtract the polynomials: \( (4x^3 - 2x^2 + 3x - 1) - (x^3 + 5x^2 - 2x + 6) \)
Definition:

Higher Degree Polynomial Subtraction: The same process applies regardless of the polynomial degree. Distribute the negative sign to all terms in the second polynomial.

Step 1: Convert to addition of the negative

\( (4x^3 - 2x^2 + 3x - 1) + (-(x^3 + 5x^2 - 2x + 6)) \)

Step 2: Distribute the negative sign

\( -(x^3 + 5x^2 - 2x + 6) = -x^3 - 5x^2 + 2x - 6 \)

Step 3: Rewrite and remove parentheses

\( 4x^3 - 2x^2 + 3x - 1 - x^3 - 5x^2 + 2x - 6 \)

Step 4: Group like terms

\( (4x^3 - x^3) + (-2x^2 - 5x^2) + (3x + 2x) + (-1 - 6) \)

Step 5: Add coefficients of like terms

\( (4 - 1)x^3 + (-2 - 5)x^2 + (3 + 2)x + (-1 - 6) \)

\( 3x^3 - 7x^2 + 5x - 7 \)

Step 6: Verify by checking each degree

3rd degree: \( 4x^3 - x^3 = 3x^3 \) ✓

2nd degree: \( -2x^2 - 5x^2 = -7x^2 \) ✓

1st degree: \( 3x - (-2x) = 3x + 2x = 5x \) ✓

Constant: \( -1 - 6 = -7 \) ✓

\( (4x^3 - 2x^2 + 3x - 1) - (x^3 + 5x^2 - 2x + 6) = 3x^3 - 7x^2 + 5x - 7 \)
Final answer:

\( 3x^3 - 7x^2 + 5x - 7 \)

Applied rules:

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

• Sign Tracking: Pay attention to double negatives

3 Subtraction with rearrangement
Exercise 3
Subtract the polynomials: \( (3x - 2x^3 + x^2 + 4) - (5x^2 - x^3 + 2x - 1) \)
Definition:

Subtraction with Non-Standard Form: When polynomials are not in standard form, it's helpful to rearrange them before subtracting to ensure proper alignment of like terms.

Step 1: Rearrange both polynomials in standard form

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

Second polynomial: \( -x^3 + 5x^2 + 2x - 1 \)

Step 2: Convert to addition of the negative

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

Step 3: Distribute the negative to the second polynomial

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

Step 4: Write the expression

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

Step 5: Group like terms

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

Step 6: Add coefficients of like terms

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

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

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

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

Applied rules:

• Standard Form: Arranging in descending order helps with organization

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

Rules and methods, laws,...
\( A(x) - B(x) = A(x) + (-B(x)) = (a_nx^n + ... + a_1x + a_0) + (-b_nx^n - ... - b_1x - b_0) \)
General Polynomial Subtraction Formula
Subtraction Rule
A - B = A + (-B)
Convert to addition
Distribution
-(a + b) = -a - b
Negative to all terms
Like Terms
Same variable parts
Combine coefficients
Standard Form
Descending order
Degrees from high to low
Key definitions:

Polynomial Subtraction: The operation of finding the difference between two polynomials 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: The process of multiplying each term inside parentheses by the factor outside.

Leading Term: The term with the highest degree.

Coefficient: The numerical factor of a term.

Steps for subtracting polynomials:
  1. Convert subtraction: Change A - B to A + (-B)
  2. Distribute negative: Multiply each term in the second polynomial by -1
  3. Remove parentheses: After distribution
  4. Group like terms: Arrange terms with the same degree together
  5. Combine like terms: Add coefficients of like terms
  6. Write in standard form: Arrange terms from highest to lowest degree
Properties: Polynomial subtraction is not commutative (A - B ≠ B - A); the result is always a polynomial; subtraction can decrease the degree of the result.
Applications: Finding differences in functions, modeling net changes, comparing mathematical models, calculating remainders or residuals.
Tip 1: Always distribute the negative sign to ALL terms in the second polynomial.
Tip 2: Double-check signs after distribution, especially for terms that were originally positive.
Tip 3: Rearrange polynomials in standard form before subtracting to avoid missing like terms.
Important rules to know:

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

• Complete Distribution: Every term in the second 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

Solution: Exercises 4 to 5
4 Subtraction with multiple variables
Exercise 4
Subtract the polynomials: \( (4x^2y - 3xy^2 + 2x - y + 5) - (x^2y + 2xy^2 - x + 3y - 2) \)
Definition:

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

Step 1: Convert to addition of the negative

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

Step 2: Distribute the negative sign

\( -(x^2y + 2xy^2 - x + 3y - 2) = -x^2y - 2xy^2 + x - 3y + 2 \)

Step 3: Rewrite the expression

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

Step 4: Group like terms

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

Step 5: Add coefficients of like terms

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

\( 3x^2y - 5xy^2 + 3x - 4y + 7 \)

Step 6: Verify by checking each term type

\( x^2y \) terms: \( 4x^2y - x^2y = 3x^2y \) ✓

\( xy^2 \) terms: \( -3xy^2 - 2xy^2 = -5xy^2 \) ✓

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

\( y \) terms: \( -y - 3y = -4y \) ✓

Constants: \( 5 - (-2) = 5 + 2 = 7 \) ✓

\( (4x^2y - 3xy^2 + 2x - y + 5) - (x^2y + 2xy^2 - x + 3y - 2) = 3x^2y - 5xy^2 + 3x - 4y + 7 \)
Final answer:

\( 3x^2y - 5xy^2 + 3x - 4y + 7 \)

Applied rules:

• Multiple Variable Like Terms: Same variables with same exponents

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

5 Real-world application
Exercise 5
The cost of producing x items is modeled by \( C(x) = 2x^2 + 3x + 100 \) dollars, and the revenue from selling x items is modeled by \( R(x) = 5x^2 + 10x + 50 \) dollars. Find the profit function \( P(x) = R(x) - C(x) \).
Definition:

Profit Function: The difference between revenue and cost functions, representing the net gain or loss from business operations.

Step 1: Set up the subtraction for the profit function

\( P(x) = R(x) - C(x) = (5x^2 + 10x + 50) - (2x^2 + 3x + 100) \)

Step 2: Convert to addition of the negative

\( P(x) = (5x^2 + 10x + 50) + (-(2x^2 + 3x + 100)) \)

Step 3: Distribute the negative sign

\( -(2x^2 + 3x + 100) = -2x^2 - 3x - 100 \)

Step 4: Write the expression

\( P(x) = 5x^2 + 10x + 50 - 2x^2 - 3x - 100 \)

Step 5: Group like terms

\( (5x^2 - 2x^2) + (10x - 3x) + (50 - 100) \)

Step 6: Add coefficients of like terms

\( (5 - 2)x^2 + (10 - 3)x + (50 - 100) \)

\( 3x^2 + 7x - 50 \)

Step 7: Interpret the result

\( P(x) = 3x^2 + 7x - 50 \) represents the profit in dollars when x items are produced and sold.

The company breaks even when P(x) = 0, makes a profit when P(x) > 0, and incurs a loss when P(x) < 0.

Profit function: \( P(x) = 3x^2 + 7x - 50 \)
Final answer:

The profit function is \( P(x) = 3x^2 + 7x - 50 \) dollars.

Applied rules:

• Business Application: Profit = Revenue - Cost

• Polynomial Subtraction: Convert to addition of the negative

• Like Terms: Combine terms with the same degree

Comprehensive Summary: Subtracting Polynomials
\( A(x) - B(x) = A(x) + (-B(x)) = (a_nx^n + ... + a_1x + a_0) + (-b_nx^n - ... - b_1x - b_0) \)
General Polynomial Subtraction Formula
Key definitions:

Polynomial Subtraction: The operation of finding the difference between two polynomials by adding the first polynomial to the negative of the second polynomial. The result is always another 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 subtracting, we effectively add the opposite coefficients.

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

Systematic approach for subtracting polynomials:
  1. Convert: Change subtraction to addition of the negative: A - B = A + (-B)
  2. Distribute: Multiply each term in the second polynomial by -1
  3. Organize: Write each polynomial in standard form
  4. Identify: Locate all like terms in both polynomials
  5. Combine: Add the coefficients of like terms
  6. Arrange: Write the result in standard form
  7. Verify: Check that all terms have been properly accounted for
Tip 1: Always distribute the negative sign to ALL terms in the second polynomial, 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 second polynomial must be multiplied by -1

• Like Terms Only: Combine terms with identical variable parts

• Coefficient Subtraction: Add coefficients while preserving variable parts

• Closure Property: The difference of polynomials is always a polynomial

• Non-Commutative: A(x) - B(x) ≠ B(x) - A(x)

Polynomial Subtraction Concepts and Techniques

Polynomial Subtraction Process

Convert to Addition

A - B = A + (-B)

→

Distribute Negative

Multiply by -1

→

Combine Like Terms

Add coefficients

Key Principles:

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

Key insight: Polynomial subtraction is essentially addition of the opposite, emphasizing the importance of proper sign handling.

Questions & Answers

Question: I always forget to distribute the negative sign to all terms. Any tips to remember this?

Answer: Here are some effective strategies to remember to distribute the negative sign:

Memory Aids:

  • Think of subtraction as "adding the opposite": \( A - B = A + (-B) \)
  • Use the phrase "every term gets flipped" when distributing the negative
  • Remember that the negative sign acts like a multiplier of -1

Visual Strategies:

  • Circle or underline each term in the second polynomial before distributing
  • Work through each term one by one: first term, second term, etc.
  • Write the distributed polynomial on a separate line before combining

Example: For \( (3x^2 + 2x - 5) - (x^2 - 4x + 3) \)

Write it as: \( (3x^2 + 2x - 5) + (-x^2 + 4x - 3) \)

Notice how each term in the second polynomial changed sign: \( x^2 \rightarrow -x^2 \), \( -4x \rightarrow +4x \), \( +3 \rightarrow -3 \).

Question: Is polynomial subtraction commutative like addition? Can I subtract in any order?

Answer: No, polynomial subtraction is NOT commutative. The order matters:

Non-Commutative Property: \( P(x) - Q(x) \neq Q(x) - P(x) \)

Why it's not commutative:

  • Subtraction is the same as adding the opposite: \( A - B = A + (-B) \)
  • But \( A + (-B) \) is generally not equal to \( B + (-A) \)
  • The result changes when you switch the order

Example:

Let \( P(x) = 2x^2 + 3x - 1 \) and \( Q(x) = x^2 - x + 4 \)

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

As you can see, the results are opposites of each other: \( P(x) - Q(x) = -(Q(x) - P(x)) \).

Always pay attention to the order of subtraction!

Question: What happens to the degree of the polynomial when I subtract? Can it change?

Answer: The degree of the resulting polynomial after subtraction can stay the same or decrease, but it will never increase:

Possible Outcomes:

  • Degree stays the same: If the leading terms don't cancel out
  • Degree decreases: If the leading terms cancel out
  • Degree cannot increase: You can only eliminate terms, not create higher-degree terms

Examples:

  • \( (3x^2 + 2x + 1) - (x^2 + x + 1) = 2x^2 + x \) (degree stays 2)
  • \( (3x^2 + 2x + 1) - (3x^2 + x + 1) = x \) (degree decreases from 2 to 1)
  • \( (3x^2 + 2x + 1) - (3x^2 + 2x + 1) = 0 \) (degree becomes undefined)

Why this happens: When subtracting polynomials, the highest-degree terms might cancel out if they have the same coefficient in both polynomials. This is different from addition, where the highest degree is typically preserved.

The degree of the difference is at most the maximum of the degrees of the original polynomials.

Question: How do I subtract polynomials with multiple variables? Is the process different?

Answer: The process is the same for polynomials with multiple variables, but you need to be more careful about identifying like terms:

Like Terms in Multiple Variables:

  • Terms are like terms if they have identical variable parts with the same exponents
  • For example: \( 3x^2y^3 \) and \( -5x^2y^3 \) are like terms
  • But \( 3x^2y^3 \) and \( 3xy^3 \) are NOT like terms

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

  • Convert to: \( (4x^2y - 3xy^2 + x) + (-2x^2y - xy^2 + 2x) \)
  • Group like terms: \( (4x^2y - 2x^2y) + (-3xy^2 - xy^2) + (x + 2x) \)
  • Combine: \( 2x^2y - 4xy^2 + 3x \)

Key Point: The variables and their exponents must match exactly for terms to be like terms. The process of distributing the negative and combining like terms remains the same.

Question: Can I subtract more than two polynomials at once? How would I do that?

Answer: Yes, you can subtract multiple polynomials. You can work left to right or group them strategically:

Method 1: Left to Right

  • Work with one subtraction at a time
  • Example: \( A - B - C = (A - B) - C \)

Method 2: Group All Negatives

  • Think of it as adding the first polynomial to the negatives of all others
  • Example: \( A - B - C = A + (-B) + (-C) \)

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

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

  • Combine like terms: \( (5x^2 - 2x^2 - x^2) + (3x + x - 2x) + (-1 - 4 + 3) \)
  • Result: \( 2x^2 + 2x - 2 \)

Tip: Grouping all negatives (Method 2) is often more efficient as it allows you to combine all like terms at once.