REDUCTION OF LIKE TERMS (WITH EXERCISES SOLVED) - MATHS - 2025

The reduction of such terms is a method used to simplify algebraic expressions. In an algebraic expression, like terms are those that have the same variable; that is, they have the same unknowns represented by a letter, and these have the same exponents.

In some cases the polynomials are extensive, and to arrive at a solution one must try to reduce the expression; This is possible when there are terms that are similar, which can be combined by applying operations and algebraic properties such as addition, subtraction, multiplication and division.

Explanation

Like terms are made up of the same variables with the same exponents, and in some cases these are only differentiated by their numerical coefficients.

Similar terms are also considered those that do not have variables; that is, those terms that only have constants. So, for example, the following are like terms:

- 6x ² - 3x ². Both terms have the same variable x ².

- 4a ² b ³ + 2a ² b ³. Both terms have the same variables a ² b ³.

- 7 - 6. The terms are constant.

Those terms that have the same variables but with different exponents are called dissimilar terms, such as:

- 9a ² b + 5ab. Variables have different exponents.

- 5x + y. The variables are different.

- b - 8. One term has a variable, the other is a constant.

Identifying the like terms that form a polynomial, these can be reduced to one, combining all those that have the same variables with the same exponents. In this way, the expression is simplified by reducing the number of terms that compose it and the calculation of its solution is facilitated.

How to do a reduction of like terms?

The reduction of like terms is done by applying the associative property of addition and the distributive property of the product. Using the following procedure, a term reduction can be done:

- First, like terms are grouped.

- The coefficients (the numbers that accompany the variables) of the like terms are added or subtracted, and the associative, commutative or distributive properties are applied, as the case may be.

- Then the new terms obtained are written, placing in front of them the sign that resulted from the operation.

Example

Reduce the terms of the following expression: 10x + 3y + 4x + 5y.

Solution

First, the terms are ordered to group those that are similar, applying the commutative property:

10x + 3y + 4x + 5y = 10x + 4x + 3y + 5y.

Then the distributive property is applied and the coefficients that accompany the variables are added to obtain the reduction of the terms:

10x + 4x + 3y + 5y

= (10 + 4) x + (3 + 5) y

= 14x + 8y.

To reduce like terms, it is important to take into account the signs of the coefficients that accompany the variable. There are three possible cases:

Reduction of like terms with equal signs

In this case the coefficients are added and the sign of the terms is placed in front of the result. Therefore, if they are positive, the resulting terms will be positive; in the case that the terms are negative, the result will have the sign (-) accompanied by the variable. For example:

a) 22ab ² + 12ab ² = 34 ab ².

b) -18x ³ - 9x ³ - 6 = -27x ³ - 6.

Reduction of like terms c

In this case, the coefficients are subtracted, and the sign of the largest coefficient is placed in front of the result. For example:

a) 15x ² y - 4x ² y + 6x ² y - 11x ² y

= (15x ² y + 6x ² y) + (- 4x ² y - 11x ² y)

= 21x ² y + (-15x ² y)

= 21x ² y - 15x ² y

= 6x ² and.

b) -5a ³ b + 3 a ³ b - 4a ³ b + a ³ b

= (3 a ³ b + a ³ b) + (-5a ³ b - 4a ³ b)

= 4a ³ b - 9a ³ b

= -5 to ³ b.

Thus, to reduce similar terms that have different signs, a single additive term is formed with all those that have a positive sign (+), the coefficients are added and the result is accompanied by the variables.

In the same way, a subtractive term is formed, with all those terms that have a negative sign (-), the coefficients are added and the result is accompanied by the variables.

Finally the sums of the two terms formed are subtracted, and the sign of the greater is placed on the result.

Reduction of like terms in operations

The reduction of like terms is an operation of algebra, which can be applied in addition, subtraction, multiplication and algebraic division.

In sums

When you have several polynomials with like terms, to reduce them, the terms of each polynomial are ordered keeping their signs, then they are written one after the other and the like terms are reduced. For example, we have the following polynomials:

3x - 4xy + 7x ² and + 5xy ².

- 6x ² y - 2xy + 9 xy ² - 8x.

In subtraction

To subtract one polynomial from another, the minuend is written and then the subtrahend with its signs changed, and then the reduction of the like terms is done. For example:

5a ³ - 3ab ² + 3b ² c

6ab ² + 2a ³ - 8b ² c

Thus, the polynomials are summarized to 3a ³ - 9ab ² + 11b ² c.

In multiplications

In a product of polynomials, the terms that make up the multiplicand are multiplied by each term that makes up the multiplier, considering that the signs of the multiplication remain the same if they are positive.

They will only be changed when multiplied by a term that is negative; that is, when two terms of the same sign are multiplied the result will be positive (+), and when they have different signs the result will be negative (-).

For example:

a) (a + b) _* (a + b)

= a ² + ab + ab + b ²

= a ² + 2ab + b ².

b) (a + b) _* (a - b)

= a ² - ab + ab - b ²

= a ² - b ².

c) (a - b) _* (a - b)

= a ² - ab - ab + b ²

= a ² - 2ab + b ².

In divisions

When you want to reduce two polynomials through a division, you must find a third polynomial that, when multiplied by the second (divisor), results in the first polynomial (dividend).

For that, the dividend and divisor terms must be ordered, from left to right, so that the variables in both are in the same order.

Then the division is performed, starting from the first term on the left of the dividend by the first term on the left of the divisor, always taking into account the signs of each term.

For example, reduce the polynomial: 10x ⁴ - 48x ³ y + 51x ² and ² + 4xy ³ - 15y ^{4 by} dividing it by the polynomial: -5x ² + 4xy + 3y ².

The resulting polynomial is -2x ² + 8xy - 5y ².

Solved exercises

First exercise

Reduce the terms of the given algebraic expression:

15a ² - 8ab + 6a ² - 6ab - 9 + 4a ² - 13 ab.

Solution

The commutative property of addition is applied, grouping the terms that have the same variables:

15a ² - 8ab + 6a ² - 6ab + 9 + 4a ² - 13

= (15a ² + 6a ² + 4a ²) + (- 8ab - 6ab) + (9 - 13).

Then the distributive property of multiplication is applied:

15a ² - 8ab + 6a ² - 6ab + 9 + 4a ² - 13

= (15 + 6 + 4) a ² + (- 8 - 6) ab + (9 - 13).

Finally, they are simplified by adding and subtracting the coefficients of each term:

15a ² - 8ab + 6a ² - 6ab + 9 + 4a ² - 13

= 25a ² - 14ab - 4.

Second exercise

Simplify the product of the following polynomials:

(8x ³ + 7xy ²) _* (8x ³ - 7 xy ²).

Solution

Each term of the first polynomial is multiplied by the second, taking into account that the signs of the terms are different; therefore, the result of its multiplication will be negative, as well as the laws of exponents must be applied.

(8x ³ + 7xy ²) _* (8x ³ - 7xy ²)

= 64 x ⁶ - 56 x ³ _* xy ² + 56 x ³ _* xy ² - 49 x ² y ⁴

= 64 x ⁶ - 49 x ² y ⁴.

References

1. Angel, AR (2007). Elementary Algebra. Pearson Education,.
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3. Jerome E. Kaufmann, KL (2011). Elementary and Intermediate Algebra: A Combined Approach. Florida: Cengage Learning.
4. Smith, SA (2000). Algebra. Pearson Education.
5. Vigil, C. (2015). Algebra and its Applications.