Associative Property

Definition

The associative property is a fundamental mathematical principle that states the order in which operations are performed does not affect the final result. It allows for the grouping of numbers or variables in an expression without changing the overall value.

5 Must Know Facts For Your Next Test

1. The associative property applies to both addition and multiplication, but not to subtraction or division.
2. In addition, the associative property states that $(a + b) + c = a + (b + c)$.
3. In multiplication, the associative property states that $(a \times b) \times c = a \times (b \times c)$.
4. The associative property allows for the rearrangement of terms in an expression without changing the final result.
5. Understanding the associative property is crucial in simplifying and solving linear equations, as well as adding and subtracting polynomials.

Review Questions

• Explain how the associative property can be applied when solving linear equations in the context of Section 2.1 'Use a General Strategy to Solve Linear Equations'.
  • The associative property is particularly useful when solving linear equations, as it allows for the rearrangement of terms without changing the overall value of the expression. For example, when solving an equation like $(2x + 3) + 4 = 7$, the associative property allows us to group the terms on the left-hand side as $(2x + 3 + 4) = 7$, simplifying the equation to $2x + 7 = 7$ and ultimately solving for $x$. This property is a key part of the general strategy for solving linear equations, as it enables us to isolate the variable and find the solution.
• Describe how the associative property can be applied when adding and subtracting polynomials in the context of Section 5.1 'Add and Subtract Polynomials'.
  • The associative property is crucial when adding and subtracting polynomials, as it allows us to group the terms in a way that simplifies the calculations. For instance, when adding the polynomials $(2x^2 + 3x - 1) + (4x^2 - 2x + 5)$, the associative property allows us to group the like terms as $(2x^2 + 4x^2) + (3x - 2x) + (-1 + 5)$, which can then be simplified to $6x^2 + x + 4$. Similarly, when subtracting polynomials, the associative property enables us to regroup the terms in a way that makes the subtraction process more efficient and accurate.
• Analyze how the associative property relates to the concept of the greatest common factor and factoring by grouping in the context of Section 6.1 'Greatest Common Factor and Factor by Grouping'.
  • The associative property plays a crucial role in the process of finding the greatest common factor (GCF) and factoring by grouping. When factoring an expression, the associative property allows us to group the terms in a way that reveals the common factors. For example, in the expression $2x^2 + 6x + 4y + 12y$, the associative property enables us to group the terms as $(2x^2 + 6x) + (4y + 12y)$, revealing the GCF of 2 and $4y$. This understanding of the associative property is essential for effectively factoring polynomials using the technique of factoring by grouping, as it allows us to identify and extract the common factors from the expression.

Related terms

Commutative Property: The commutative property states that the order of the operands in an addition or multiplication expression does not affect the result.

Distributive Property: The distributive property allows for the distribution of a single factor over the terms of an expression, simplifying calculations.

Identity Property: The identity property states that adding 0 to a number or multiplying a number by 1 does not change the original value.