The distributive property is a fundamental algebraic rule that states that the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions involving multiplication and addition or subtraction.
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The distributive property is used to simplify expressions involving multiplication and addition or subtraction, such as $a(b + c) = ab + ac$.
The distributive property is a key concept in solving linear equations, as it allows for the isolation of variables.
The distributive property is essential for adding and subtracting polynomials, as it allows for the combination of like terms.
The distributive property is used to factor polynomials, particularly trinomials, by identifying the greatest common factor.
The distributive property is also applicable in the context of rational expressions, where it is used to simplify products and quotients.
Review Questions
Explain how the distributive property can be used to solve linear equations.
The distributive property is crucial in solving linear equations because it allows for the isolation of variables. By applying the distributive property, terms with variables can be separated from constant terms, enabling the equation to be simplified and solved for the unknown variable. For example, in the equation $3(x + 5) = 21$, the distributive property can be used to rewrite the left-hand side as $3x + 15 = 21$, making it easier to solve for $x$.
Describe how the distributive property is used in the context of polynomial operations.
The distributive property is essential for adding and subtracting polynomials, as it allows for the combination of like terms. When adding or subtracting polynomials, the distributive property is used to distribute the coefficients to each term within the polynomial. For instance, in the expression $(2x + 3) + (4x - 5)$, the distributive property can be applied to simplify the expression as $2x + 3 + 4x - 5 = 6x - 2$. Additionally, the distributive property is used in the factorization of polynomials, particularly trinomials, by identifying the greatest common factor.
Analyze how the distributive property can be applied in the context of rational expressions.
The distributive property is also applicable in the realm of rational expressions, where it is used to simplify products and quotients. When multiplying or dividing rational expressions, the distributive property allows for the distribution of the numerator and denominator terms. For example, in the expression $\frac{2x(3x + 5)}{x(x + 1)}$, the distributive property can be used to rewrite the numerator as $6x^2 + 10x$, resulting in the simplified expression $\frac{6x^2 + 10x}{x(x + 1)}$. This simplification process is crucial in working with rational expressions and their operations.