Question

Find the product.\((3 x-5)(2 x+1)\)

Short answer

The product of \((3x-5)(2x+1)\) is \(6x^2 - 7x - 5\).

Step-by-step solution

Distributing first term of the first binomial

Multiply the first term of the first binomial \((3x)\) by every term in the second binomial \((2x+1)\). This results in \(6x^2 + 3x\).

Distributing second term of the first binomial

Multiply the second term of the first binomial \((-5)\) by every term in the second binomial \((2x + 1)\). This results in \(-10x - 5\).

Combine like terms

Now add the result from step 1 and step 2 to get the final expression. So, \(6x^2 + 3x - 10x - 5\) simplifies to \(6x^2 - 7x - 5\).

Key concepts

Distributive Property

Understanding the distributive property is crucial when working with binomial multiplication. This property allows us to multiply a single term by each term within a parenthesis. For instance, if we have an expression like \(a(b + c)\), the distributive property enables us to multiply \(a\) with \(b\) and \(c\) separately, resulting in \(ab + ac\).

In the context of our exercise \( (3x - 5)(2x + 1) \), we apply the distributive property twice. First, we distribute \(3x\) across the terms \(2x\) and \(1\), yielding \(6x^2 + 3x\). Next, we distribute \( -5\) in the same manner, resulting in \( -10x - 5\). This property makes it possible to expand binomials into a form that can then be simplified further.

Combining Like Terms

Once the distributive property has been applied, we often end up with an expression that contains like terms. Like terms are terms that have the same variables raised to the same power. These terms can be combined - added or subtracted - to simplify the expression further.

In our given problem after applying the distributive property, we obtain \(6x^2 + 3x - 10x - 5\). Here, \(3x\) and \( -10x\) are like terms since they both contain the variable \(x\) raised to the first power. To combine them, we add their coefficients (the numbers in front of the variable), yielding \( -7x\). This simplification results in the final, more streamlined expression \(6x^2 - 7x - 5\).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operators (such as \(+\) and \( - \)). Algebraic expressions can be simple, with just one term, or they can be polynomials with many terms.

In the exercise concerning binomial multiplication, the product of the binomials results in a polynomial with three terms: \(6x^2 - 7x - 5\). Each of these terms is a part of our final algebraic expression, which represents numbers in an abstract and general form, allowing for a wide range of values to be substituted for the variables. Grasping the concept of algebraic expressions is foundational in algebra and enables students to move on to more complex mathematical equations.