Collecting Like Terms

In algebra, we often need to simplify expressions by combining like terms. Like terms are terms that have the same variable(s) raised to the same power.

Remember, we cannot add or subtract unlike terms. For example, the expression a + ab remains as it is because a and ab are unlike terms. We cannot simplify any further.

Similarly, the expression a + { a }^{ 2 } + { a }^{ 3 } remains as it is because a, { a }^{ 2 }, { a }^{ 3 } are all unlike terms.

Let’s look at some examples of like and unlike terms:

Like terms: a, 2a, 3a, -7a, 100a

Unlike terms: 5a, 5b and 2a, 3b

When simplifying expressions, we collect the like terms together, then perform addition and subtraction as needed.

Let’s look at some examples:

Examples

Example 1:

Simplify: 2a + 3ab - a - ab

Combine like terms:

2a - a + 3ab - ab = a + 2ab2a - a + 3ab - ab = a + 2ab

Example 2:

Simplify: 2a + a^{2} + 3a + 5a^{2}

Combine like terms:

2a + 3a + a^{2} + 5a^{2} = 5a + 6a^{2}2a + 3a + a^{2} + 5a^{2} = 5a + 6a^{2}

Example 3:

Simplify: 3ab - abc + 5ab - 2abc

Combine like terms:

3ab + 5ab - abc - 2abc = 8ab - 3abc3ab + 5ab - abc - 2abc = 8ab - 3abc

Example 4:

Simplify: x^{2} + 5y^{2} + 3x^{2} - y^{2}

Combine like terms:

4x^{2} + 4y^{2}4x^{2} + 4y^{2}

Example 5:

Simplify: a + a^{2} + a^{3} + b + b^{2} + b^{3} - 2a^{2} - 3b^{3}

Combine like terms:

a + (1 - 2)a^{2} + a^{3} + b + b^{2} + (1 - 3)b^{3}a + (1 - 2)a^{2} + a^{3} + b + b^{2} + (1 - 3)b^{3}

Since there are no other like terms to combine, the simplified expression is:

a - a^{2} + a^{3} + b + b^{2} - 2b^{3}a - a^{2} + a^{3} + b + b^{2} - 2b^{3}

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