Multiply $(3 x-5)(x+y)$
a. $4 x-3 x y-y$
b. $3 x^2+3 x y-5 x-5 y$
C. $3 x+x y-5 x-5 y$
d. $3 x^2+3 y-5 x-5 y$
Answer (1)
Apply the distributive property to multiply each term in the first binomial by each term in the second binomial.
Multiply 3 x by x and y to get 3 x 2 + 3 x y .
Multiply − 5 by x and y to get − 5 x − 5 y .
Combine the terms to get the final expression: 3 x 2 + 3 x y − 5 x − 5 y .
Explanation
Understanding the Problem We are given the expression ( 3 x − 5 ) ( x + y ) to expand. Our goal is to multiply these two binomials and simplify the result.
Applying the Distributive Property We will use the distributive property (also known as the FOIL method) to expand the product of the two binomials. This means we multiply each term in the first binomial by each term in the second binomial.
Multiplying the First Term First, we multiply 3 x by both x and y :
3 x × x + 3 x × y = 3 x 2 + 3 x y
Multiplying the Second Term Next, we multiply − 5 by both x and y :
− 5 × x + ( − 5 ) × y = − 5 x − 5 y
Combining the Terms Now, we combine the terms we obtained in the previous steps: 3 x 2 + 3 x y − 5 x − 5 y
Final Simplification and Answer We check if the resulting expression can be further simplified by combining like terms. In this case, there are no like terms to combine, so the expression is already in its simplest form. Comparing our final expression 3 x 2 + 3 x y − 5 x − 5 y with the given options, we can see that it matches option b.
Examples
Understanding how to expand binomials like ( 3 x − 5 ) ( x + y ) is useful in many areas, such as calculating areas and volumes in geometry, modeling growth and decay in science, and even in financial calculations. For example, if you're designing a rectangular garden where one side is ( 3 x − 5 ) meters and the other is ( x + y ) meters, the expanded form 3 x 2 + 3 x y − 5 x − 5 y would give you the total area of the garden in terms of x and y . This skill is also crucial in simplifying more complex algebraic expressions and solving equations.
