1. Adding and Subtracting Polynomials

Adding and Subtracting Polynomials

Introduction

Polynomials are mathematical expressions made up of numbers, variables, and whole-number exponents. They are used throughout algebra to represent patterns, describe relationships, and solve problems involving quantities that change.

For example:

3x + 5

2x² − 4x + 7

5a³ + 2a² − a + 6

We can add and subtract polynomials by combining terms that have the same variables raised to the same powers.

Consider:

(3x + 4) + (2x + 5)

We can combine the x terms and the constant terms:

3x + 2x + 4 + 5

= 5x + 9

The key to simplifying polynomial expressions is understanding like terms, paying attention to positive and negative signs, and following the correct order of operations.

 

 

 

 

 

 

 

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What Is a Polynomial?

A polynomial is an expression consisting of terms that contain constants, variables, and nonnegative integer exponents.

Examples include:

• 4x + 3

• 2x² − 5x + 1

• 7a³ + 3a² − 2a + 9

• 5

• −3y⁴ + 2y

Each polynomial consists of one or more terms separated by addition or subtraction signs.

Parts of a Polynomial

Consider:

5x² − 3x + 7

This polynomial contains three terms.

A coefficient is the numerical factor multiplying a variable.

A constant is a term without a variable.

An exponent indicates how many times a variable is multiplied by itself.

For example:

x³ = x × x × x

Polynomial Vocabulary

Polynomials are sometimes classified by their number of terms.

A polynomial with more than three terms is generally called a polynomial.

Understanding Like Terms

Like terms are terms that have exactly the same variable parts, including the same exponents.

For example:

3x and 7x are like terms.

5x² and −2x² are like terms.

4ab and 9ab are like terms.

However:

3x and 3x² are not like terms.

5x and 5y are not like terms.

2ab and 2a²b are not like terms.

Why Must the Variables and Exponents Match?

Consider:

3 apples + 5 apples = 8 apples

But:

3 apples + 5 oranges cannot be combined into 8 apples or 8 oranges.

Algebra works similarly.

We can combine:

3x + 5x = 8x

But we cannot combine:

3x + 5x²

into a single term because x and x² represent different variable parts.

Identifying Like Terms

Important: All constant terms are like terms because they contain no variables.

Combining Like Terms

To combine like terms, add or subtract their coefficients while keeping the variable part unchanged.

Example

Simplify:

4x + 7x

Add the coefficients:

4 + 7 = 11

Keep the variable:

4x + 7x = 11x

Another Example

Simplify:

9x² − 4x²

Subtract the coefficients:

9 − 4 = 5

Keep x²:

9x² − 4x² = 5x²

Combining Negative Terms

Simplify:

−3y − 5y

Add the negative coefficients:

−3 + (−5) = −8

Therefore:

−3y − 5y = −8y

Combining Positive and Negative Terms

Simplify:

8a − 12a

8 − 12 = −4

Therefore:

8a − 12a = −4a

Simplifying Polynomials with Several Terms

A polynomial may contain several groups of like terms.

Worked Example

Simplify:

5x + 3 + 2x + 7

Group like terms:

(5x + 2x) + (3 + 7)

Combine:

7x + 10

Final answer: 7x + 10

Worked Example with Different Powers

Simplify:

4x² + 3x − 2x² + 5x + 6

Group like terms:

(4x² − 2x²) + (3x + 5x) + 6

Combine:

2x² + 8x + 6

Final answer: 2x² + 8x + 6

Notice that the x² terms and x terms are combined separately.

Arranging Polynomials in Standard Form

A polynomial is in standard form when its terms are arranged from the highest exponent to the lowest exponent.

For example:

3x³ + 5x² − 2x + 7

is in standard form.

The highest power appears first.

Worked Example

Write the following expression in standard form:

4 + 3x − 2x³ + 5x²

Arrange by descending powers:

−2x³ + 5x² + 3x + 4

Why Use Standard Form?

Standard form makes it easier to:

• Identify like terms.

• Compare polynomials.

• Add and subtract expressions.

• Identify the degree of a polynomial.

• Check whether terms have been omitted.

Degree of a Polynomial

The degree of a nonzero polynomial in one variable is its highest exponent with a nonzero coefficient.

For example:

5x³ + 2x² − 7

has degree 3.

The polynomial:

4x⁵ − 3x + 8

has degree 5.

Adding Polynomials

Adding polynomials involves combining all like terms from the expressions.

General Method

1. Write the polynomials inside parentheses.

2. Remove the parentheses when they are preceded by a plus sign.

3. Group like terms.

4. Add their coefficients.

5. Write the result in standard form.

Worked Example

Add:

(3x + 5) + (2x + 7)

Remove the parentheses:

3x + 5 + 2x + 7

Group like terms:

(3x + 2x) + (5 + 7)

Simplify:

5x + 12

Important Rule

When adding polynomials, a plus sign before parentheses does not change the signs of the terms inside.

For example:

+(4x − 3) = 4x − 3

Adding Polynomials with Quadratic Terms

Worked Example

Simplify:

(3x² + 4x + 2) + (5x² − 2x + 7)

Remove the parentheses:

3x² + 4x + 2 + 5x² − 2x + 7

Group like terms:

(3x² + 5x²) + (4x − 2x) + (2 + 7)

Combine:

8x² + 2x + 9

Final answer: 8x² + 2x + 9

Visualising Polynomial Addition

Algebra tiles can represent polynomial terms using shapes of different sizes.

 

 

 

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For example:

3x² + 2x + 4

can be represented using:

• Three x² tiles.

• Two x tiles.

• Four unit tiles.

Adding:

2x² + 3x + 1

produces:

Five x² tiles, five x tiles, and five unit tiles.

Therefore:

(3x² + 2x + 4) + (2x² + 3x + 1) = 5x² + 5x + 5

Adding Polynomials Vertically

Polynomials can also be added using a vertical arrangement.

This method is particularly useful for expressions containing many terms.

Worked Example

Add:

4x² + 3x + 7

and:

2x² + 5x − 4

Arrange like terms in columns:

   4x² + 3x + 7
+  2x² + 5x − 4
----------------
   6x² + 8x + 3

The result is:

6x² + 8x + 3

Why Is Vertical Addition Useful?

It helps prevent mistakes by keeping:

• x² terms together.

• x terms together.

• Constant terms together.

This becomes especially useful for higher-degree polynomials.

Adding Polynomials with Missing Terms

Some polynomials do not contain every power of the variable.

Worked Example

Add:

(5x³ + 2x − 4) + (3x³ + 6x² − 7x + 9)

Notice that the first polynomial has no x² term.

Group like terms:

(5x³ + 3x³) + 6x² + (2x − 7x) + (−4 + 9)

Simplify:

8x³ + 6x² − 5x + 5

Missing terms can be treated as having a coefficient of zero.

For example:

5x³ + 2x − 4

can be written as:

5x³ + 0x² + 2x − 4

Subtracting Polynomials

Subtracting polynomials requires special attention to negative signs.

When a polynomial is subtracted, the sign of every term inside its parentheses must change.

Important Rule

−(a + b) = −a − b

−(a − b) = −a + b

This is an application of the distributive property.

Worked Example

Subtract:

(5x + 8) − (2x + 3)

Distribute the negative sign:

5x + 8 − 2x − 3

Group like terms:

(5x − 2x) + (8 − 3)

Simplify:

3x + 5

Subtracting Polynomials with Negative Terms

Worked Example

Simplify:

(7x − 4) − (3x − 9)

Distribute the negative sign:

7x − 4 − 3x + 9

Notice that −9 becomes +9 when the second polynomial is subtracted.

Group:

(7x − 3x) + (−4 + 9)

Simplify:

4x + 5

Another Example

Simplify:

(4x² + 6x − 3) − (2x² − 5x + 8)

Distribute the negative sign:

4x² + 6x − 3 − 2x² + 5x − 8

Group like terms:

(4x² − 2x²) + (6x + 5x) + (−3 − 8)

Simplify:

2x² + 11x − 11

Understanding Why Signs Change

Subtracting a polynomial is equivalent to adding its opposite.

For example:

A − B = A + (−B)

If:

B = 3x − 5

then:

−B = −3x + 5

Therefore:

(7x + 2) − (3x − 5)

becomes:

(7x + 2) + (−3x + 5)

= 4x + 7

This is why every term in the subtracted polynomial changes sign.

Subtracting Polynomials Vertically

Worked Example

Subtract:

(3x² + 2x − 5)

from:

(8x² − 4x + 7)

Write the subtraction:

(8x² − 4x + 7) − (3x² + 2x − 5)

Arrange vertically:

    8x² − 4x + 7
− ( 3x² + 2x − 5 )
------------------
    5x² − 6x + 12

Calculate each column:

8x² − 3x² = 5x²

−4x − 2x = −6x

7 − (−5) = 12

Therefore:

5x² − 6x + 12

Adding and Subtracting Polynomials with Multiple Variables

Polynomials can contain more than one variable.

The same rules apply.

Only terms with identical variable parts can be combined.

Worked Example

Simplify:

(3a + 4b) + (5a − 2b)

Remove parentheses:

3a + 4b + 5a − 2b

Group like terms:

(3a + 5a) + (4b − 2b)

Simplify:

8a + 2b

Worked Example with Products of Variables

Simplify:

(4xy + 3x² − 2y) − (2xy − 5x² + 6y)

Distribute the negative sign:

4xy + 3x² − 2y − 2xy + 5x² − 6y

Group like terms:

(3x² + 5x²) + (4xy − 2xy) + (−2y − 6y)

Simplify:

8x² + 2xy − 8y

Important Reminder

The terms xy and x²y are not like terms.

The terms x²y and yx² are like terms because multiplication is commutative.

Simplifying Expressions with Multiple Operations

Some polynomial expressions contain several additions and subtractions.

The safest approach is to remove parentheses carefully and combine like terms.

Worked Example

Simplify:

(3x² + 2x − 4) + (5x² − 7x + 6) − (2x² + x − 3)

Remove the first two sets of parentheses:

3x² + 2x − 4 + 5x² − 7x + 6

Distribute the negative sign across the final polynomial:

−2x² − x + 3

Combine the complete expression:

3x² + 2x − 4 + 5x² − 7x + 6 − 2x² − x + 3

Group like terms:

(3x² + 5x² − 2x²) + (2x − 7x − x) + (−4 + 6 + 3)

Simplify:

6x² − 6x + 5

Using the Distributive Property Before Combining Terms

Some expressions contain multiplication as well as addition or subtraction.

Multiplication must be completed before like terms are combined.

Worked Example

Simplify:

3(2x + 4) + 2(x − 5)

Distribute:

3(2x + 4) = 6x + 12

2(x − 5) = 2x − 10

Combine:

6x + 12 + 2x − 10

= 8x + 2

Final answer: 8x + 2

Worked Example with a Negative Multiplier

Simplify:

4(x + 3) − 2(3x − 5)

Distribute:

4x + 12 − 6x + 10

Combine:

(4x − 6x) + (12 + 10)

= −2x + 22

Important Observation

The negative multiplier −2 affects both terms:

−2(3x − 5) = −6x + 10

Simplifying Nested Parentheses

Expressions may contain parentheses inside other parentheses.

Worked Example

Simplify:

2x + [3x − (4x − 5)]

Begin with the inner parentheses:

−(4x − 5) = −4x + 5

Substitute:

2x + [3x − 4x + 5]

Simplify inside the brackets:

2x + [−x + 5]

Remove brackets:

2x − x + 5

= x + 5

Another Example

Simplify:

3[2x − (x − 4)] − 5

Inside the brackets:

2x − (x − 4)

= 2x − x + 4

= x + 4

Now multiply:

3(x + 4) − 5

= 3x + 12 − 5

= 3x + 7

Simplifying Higher-Degree Polynomials

The same principles apply to cubic and higher-degree polynomials.

Worked Example

Simplify:

(4x³ − 2x² + 5x − 7) + (3x³ + 6x² − 8x + 4)

Group like terms:

(4x³ + 3x³) + (−2x² + 6x²) + (5x − 8x) + (−7 + 4)

Simplify:

7x³ + 4x² − 3x − 3

Worked Example: Higher-Degree Subtraction

Simplify:

(6x⁴ + 3x³ − 2x + 5) − (2x⁴ − 4x³ + 7x² + x − 8)

Distribute the negative sign:

6x⁴ + 3x³ − 2x + 5 − 2x⁴ + 4x³ − 7x² − x + 8

Group like terms:

(6x⁴ − 2x⁴) + (3x³ + 4x³) − 7x² + (−2x − x) + (5 + 8)

Simplify:

4x⁴ + 7x³ − 7x² − 3x + 13

Checking Polynomial Simplification

A useful way to check polynomial addition and subtraction is to substitute a numerical value for the variable.

The original and simplified expressions should produce the same result.

Worked Example

Suppose we simplify:

(3x + 4) + (2x − 1)

to:

5x + 3

Check using x = 2.

Original Expression

(3(2) + 4) + (2(2) − 1)

= (6 + 4) + (4 − 1)

= 10 + 3

= 13

Simplified Expression

5(2) + 3

= 10 + 3

= 13

Both expressions give the same result.

This supports the simplification.

Important Limitation

Checking one numerical value does not prove that two expressions are equivalent for every possible value.

However, it is a useful method for detecting many common errors.

Interactive Practice: Combining Like Terms

Change the coefficients to investigate how polynomial addition works.

Real-World Application: Perimeter of a Rectangle

Polynomials can represent dimensions of geometric shapes.

Suppose a rectangle has:

Length = 3x + 5

Width = 2x + 1

The perimeter of a rectangle is:

P = 2L + 2W

Substitute:

P = 2(3x + 5) + 2(2x + 1)

Distribute:

P = 6x + 10 + 4x + 2

Combine like terms:

P = 10x + 12

The perimeter is represented by the polynomial 10x + 12.

 

 

 

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Extension

If x = 4 cm:

P = 10(4) + 12

= 40 + 12

P = 52 cm

Real-World Application: Comparing Costs

Suppose two companies offer different pricing plans.

Company A:

A(x) = 5x + 20

Company B:

B(x) = 3x + 35

Here, x represents the number of units purchased, and costs are measured in dollars.

Find the difference:

A(x) − B(x)

Substitute:

(5x + 20) − (3x + 35)

Distribute the negative sign:

5x + 20 − 3x − 35

Combine like terms:

A(x) − B(x) = 2x − 15

Interpretation

The expression 2x − 15 represents how much more Company A costs than Company B.

If x = 10:

2(10) − 15 = 5

Company A costs $5 more.

If x = 5:

2(5) − 15 = −5

Company A costs $5 less.

Polynomial subtraction helps us compare changing quantities.

Real-World Application: Combining Areas

Suppose two regions have areas represented by:

A₁ = 3x² + 2x + 5

A₂ = 2x² + 4x + 3

The combined area is:

A_total = A₁ + A₂

Substitute:

(3x² + 2x + 5) + (2x² + 4x + 3)

Combine like terms:

A_total = 5x² + 6x + 8

This type of calculation is useful in geometry, architecture, and engineering.

Common Mistakes

Combining unlike terms

Incorrect:

3x + 4x² = 7x³

Correct:

3x + 4x² cannot be combined into a single term.

Adding exponents when adding like terms

Incorrect:

2x² + 3x² = 5x⁴

Correct:

2x² + 3x² = 5x²

Exponents are not added when combining like terms.

Forgetting to change every sign when subtracting

Incorrect:

(5x + 3) − (2x − 4) = 5x + 3 − 2x − 4

Correct:

5x + 3 − 2x + 4

Ignoring negative coefficients

Incorrect:

4x − 9x = 5x

Correct:

4x − 9x = −5x

Combining constants with variable terms

Incorrect:

3x + 5 = 8x

Correct:

3x + 5 is already simplified.

Forgetting to distribute multiplication

Incorrect:

3(x + 4) = 3x + 4

Correct:

3(x + 4) = 3x + 12

Failing to arrange terms clearly

Writing expressions in standard form makes it easier to identify errors.

Dropping terms during simplification

Every original term must be accounted for when parentheses are removed.

Confusing subtraction order

A − B is generally not equal to B − A.

Key Terms

• Polynomial: Algebraic expression containing terms with nonnegative integer powers of variables.

• Term: Individual part of an algebraic expression separated by addition or subtraction.

• Coefficient: Numerical factor multiplying a variable.

• Variable: Symbol representing a number or quantity.

• Constant: Term without a variable.

• Exponent: Number indicating the power of a variable.

• Like Terms: Terms with identical variable parts and exponents.

• Unlike Terms: Terms with different variable parts or exponents.

• Simplify: Rewrite an expression in an equivalent, usually more compact form.

• Equivalent Expressions: Expressions that have the same value for every permitted variable value.

• Monomial: Polynomial with one term.

• Binomial: Polynomial with two terms.

• Trinomial: Polynomial with three terms.

• Degree: Highest exponent in a nonzero polynomial in one variable.

• Standard Form: Polynomial written in descending order of powers.

• Distributive Property: Property used to multiply a factor across terms inside parentheses.

• Additive Inverse: Number or expression that adds to another to produce zero.

• Polynomial Addition: Combining corresponding like terms from polynomials.

• Polynomial Subtraction: Adding the opposite of a polynomial.

• Substitution: Replacing a variable with a numerical value or another expression.

• Verification: Checking whether a result is correct.

Key Takeaways

• Polynomials consist of terms containing coefficients, variables, and whole-number exponents.

• Like terms must have exactly the same variable parts and exponents.

• Only like terms can be combined through addition or subtraction.

• When combining like terms, add or subtract coefficients and keep the variable part unchanged.

• Polynomial addition involves combining corresponding like terms.

• When adding polynomials, a plus sign before parentheses does not change the signs inside.

• When subtracting polynomials, change the sign of every term in the polynomial being subtracted.

• Subtraction can be understood as adding the opposite.

• The distributive property must be applied before combining terms when multiplication is present.

• Standard form arranges polynomial terms from highest to lowest degree.

• Vertical alignment can help simplify longer polynomial expressions.

• Polynomials with multiple variables follow the same rules.

• Substituting numerical values can help check simplification.

• Polynomial operations are useful in geometry, finance, science, and algebraic modelling.

Check Your Understanding

Simplify each expression.

Identifying and Combining Like Terms

1. 3x + 5x

2. 8a − 3a

3. 4x² + 7x²

4. 6y − 10y

5. 3x² + 5x − x² + 2x

Adding Polynomials

6. (2x + 3) + (5x + 7)

7. (4x − 5) + (3x + 9)

8. (3x² + 2x + 1) + (4x² − x + 6)

9. (5a² − 3a + 4) + (2a² + 6a − 7)

10. (2x³ + 4x² − 5) + (3x³ − 2x² + 7)

Subtracting Polynomials

11. (7x + 5) − (3x + 2)

12. (8x − 4) − (5x − 9)

13. (6x² + 3x − 2) − (2x² − 4x + 5)

14. (9a² − 5a + 8) − (4a² + 2a − 6)

15. (5x³ + 2x² − 3x + 4) − (2x³ − 4x² + x − 7)

Multiple Operations

16. 3(x + 4) + 2(x − 5)

17. 4(2x − 3) − 3(x + 2)

18. (3x² + 2x − 1) + (2x² − 5x + 4) − (x² + x − 3)

19. 2[3x − (x − 4)] + 5

20. 3(2x² − x + 4) − 2(x² + 3x − 5)

Answers

Extended Challenge

Combining Like Terms

1. Simplify: 7x² − 4x + 3x² + 9x − 6.

2. Simplify: 5a²b + 3ab² − 2a²b + 7ab².

3. Simplify: 8x³ − 5x² + 4x − 3x³ + 7x² − 9x.

4. Write 5 + 3x² − 7x + 2x³ in standard form.

5. Explain why 3xy and 5yx are like terms.

Adding Polynomials

6. (4x² + 7x − 3) + (6x² − 2x + 9)

7. (3x³ − 5x² + 2x − 7) + (4x³ + 2x² − 6x + 10)

8. (5a² + 3ab − 2b²) + (2a² − 7ab + 4b²)

9. (2x⁴ − 3x³ + 5x − 8) + (4x⁴ + 6x³ − 2x² + 7)

10. Create two different quadratic polynomials whose sum is 8x² + 3x − 5.

Subtracting Polynomials

11. (8x² − 3x + 7) − (5x² + 4x − 9)

12. (6x³ + 2x² − 5x + 3) − (4x³ − 7x² + 2x − 8)

13. (7a² − 4ab + 3b²) − (2a² + 6ab − 5b²)

14. (9x⁴ − 3x³ + 5x² − 2) − (4x⁴ + 2x³ − 7x + 6)

15. Explain why subtracting a polynomial requires changing the signs of all its terms.

Multiple Operations

16. 3(2x + 5) − 4(x − 3)

17. 2(3x² − 4x + 5) + 3(x² + 2x − 7)

18. 4[2x − (3x − 5)] + 3(x − 2)

19. (5x² + 3x − 4) − 2(x² − 5x + 7) + (3x² − x + 6)

20. 3[2x² − (x² − 4x + 3)] − 2(x² + x − 5)

Final Challenge: Polynomial Operations Investigation

A community centre is planning two rectangular garden areas.

Garden A has dimensions:

Length = 3x + 4 metres

Width = x + 2 metres

Garden B has dimensions:

Length = 2x + 5 metres

Width = x + 3 metres

Complete the following investigation.

1. Write an expression for the perimeter of Garden A.

2. Simplify the perimeter expression.

3. Write an expression for the perimeter of Garden B.

4. Simplify the perimeter expression.

5. Write a polynomial representing the combined perimeter of both gardens.

6. Simplify the combined perimeter.

7. Subtract the perimeter of Garden B from the perimeter of Garden A.

8. Simplify the difference.

9. Explain what a positive result means.

10. Explain what a negative result means.

11. Evaluate both perimeter expressions when x = 4.

12. Determine which garden has the greater perimeter when x = 4.

13. Find the difference between their perimeters.

14. Explain why like terms must be combined when simplifying the expressions.

15. Verify your results using direct substitution.

16. Write a short explanation of how polynomial addition and subtraction help solve geometric problems.

Further Investigation

Create two different polynomial expressions representing real-world quantities.

Your expressions might represent costs, distances, areas, populations, or quantities produced.

Add the expressions and interpret the result.

Subtract one expression from the other and explain what the difference represents.

Finally, substitute a realistic numerical value for the variable and verify your calculations.