Question

Use the Distributive Property to remove the parentheses. $$ (3 x-1)(x+5) $$

Short answer

3x^2 + 14x - 5

Step-by-step solution

Apply the Distributive Property

The Distributive Property states that for any numbers a, b, and c, the equation a(b+c) = ab + ac. Use this property to expand each term in the binomial (3x - 1) with each term in the binomial (x + 5).

Multiply each term individually

First, multiply 3x by x to get 3x^2.Next, multiply 3x by 5 to get 15x.Then, multiply -1 by x to get -1x.Finally, multiply -1 by 5 to get -5.

Combine like terms

Now combine all the terms: 3x^2 + 15x - 1x - 5. Next, simplify by combining the like terms: 15x and -1x to get 14x.

Write the simplified expression

The simplified expression becomes 3x^2 + 14x - 5.

Key concepts

Expanding Binomials

Expanding binomials is a fundamental concept in algebra. It involves breaking down and distributing each term of one binomial across the terms of another binomial. In our problem, we start with \[\begin{equation}(3x-1)(x+5)otag\balance\end{equation}\] To expand this expression, we distribute each term in the first binomial across each term in the second binomial:

• Multiply 3x by x to get 3x².
• Then, multiply 3x by 5 to get 15x.
• Next, multiply -1 by x to get -1x.
• Finally, multiply -1 by 5 to get -5.

Once we perform all these multiplications, we get: \[\begin{equation}3x^2 + 15x - 1x - 5otag\balance\end{equation}\]. Always ensure to keep the signs in front of the numbers accurate to avoid mistakes.

Like Terms

Like terms in algebra are terms that have the same variables raised to the same power. Combining like terms is crucial to simplifying expressions. In our example, after expanding the binomials, we get:\[\begin{equation}3x^2 + 15x - 1x - 5.otag\balance\end{equation}\] Out of these terms, 15x and -1x are like terms because they both have the variable ‘x’ raised to the same power.

• We combine 15x and -1x to get 14x.
• Other terms like 3x² and -5 remain unchanged.

Combining like terms simplifies our expression and makes it easier to understand.

Simplifying Expressions

Simplifying expressions is a way to make them more manageable and easier to work with. This often involves combining like terms. From our example, after expanding and identifying like terms, we have:\[\begin{equation}3x^2 + 15x - 1x - 5.otag\balance\end{equation}\] We then combine the like terms, specifically 15x and -1x, to get:\[\begin{equation}3x^2 + 14x - 5.otag\balance\end{equation}\] Simplifying expressions can reduce the complexity of equations and make further calculations more straightforward. This is an essential skill in algebra and higher-level mathematics.

Multiplication of Polynomials

The multiplication of polynomials involves using the distributive property to expand the product. Polynomials are algebraic expressions that include coefficients and variables. To multiply them, each term in one polynomial is multiplied by each term in the other polynomial. In our problem \[\begin{equation}(3x-1)(x+5),otag\end{equation}\] we used the distributive property to multiply each term:

• 3x multiplied by x and by 5.
• -1 multiplied by x and by 5.

Combining all these products, we expanded the binomials into a polynomial expression:\[\begin{equation}3x^2 + 15x - 1x - 5.otag\balance\end{equation}\] Then, by combining like terms, we simplified the final expression to:\[\begin{equation}3x^2 + 14x - 5.otag\balance\end{equation}\] Understanding the multiplication of polynomials is a building block for more advanced algebra topics.