The distributive property is a fundamental algebraic principle that allows for the simplification of expressions involving multiplication. It states that the product of a number and a sum is equal to the sum of the individual products of the number with each addend.
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The distributive property is used to simplify expressions involving the multiplication of a term with a sum, such as $a(b + c) = ab + ac$.
The distributive property is a crucial concept in solving linear equations, as it allows for the isolation of variables on one side of the equation.
The distributive property is also essential in the multiplication and division of polynomials, as it enables the expansion and factorization of polynomial expressions.
The distributive property is one of the fundamental properties of real numbers, along with the commutative and associative properties.
The distributive property is often used in combination with other algebraic properties, such as the commutative and associative properties, to simplify complex expressions.
Review Questions
Explain how the distributive property can be used to simplify the expression $3(x + 5)$.
To simplify the expression $3(x + 5)$ using the distributive property, we multiply each term inside the parentheses by the number outside the parentheses. This gives us $3x + 15$. The distributive property allows us to distribute the 3 to both the $x$ and the 5, resulting in the simplified expression.
Describe how the distributive property can be used to solve the equation $2(x - 3) = 14$.
To solve the equation $2(x - 3) = 14$ using the distributive property, we first distribute the 2 to the $x$ and the -3 inside the parentheses. This gives us $2x - 6 = 14$. We can then isolate the variable $x$ by adding 6 to both sides, resulting in $2x = 20$. Finally, we divide both sides by 2 to solve for $x$, which gives us $x = 10$. The distributive property allows us to simplify the equation and isolate the variable.
Analyze how the distributive property is used in the multiplication of polynomials, such as $(2x + 3)(4x - 1)$.
When multiplying polynomials, such as $(2x + 3)(4x - 1)$, the distributive property is used to expand the expression. First, we distribute the $(4x - 1)$ to each term in the $(2x + 3)$ expression, resulting in $8x^2 - 2x + 12x - 3$. Then, we combine like terms, which gives us $8x^2 + 10x - 3$. The distributive property allows us to break down the multiplication of the two polynomials into a series of simpler multiplications, leading to the final expanded expression.