Question

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

Short answer

The product of the two binomials (5x - 1) and (5x + 2) is \(25x^2 + 5x - 2\).

Step-by-step solution

Distribute the First Term of the First Binomial

Multiply the first term of the first binomial (5x) with both terms of the second binomial. It will be: 5x * 5x + 5x * 2. The results are 25x^2 and 10x, respectively.

Distribute the Second Term of the First Binomial

Multiply the second term of the first binomial (-1) with both terms of the second binomial. It will be: -1 * 5x + -1 * 2. The results are -5x and -2, respectively.

Sum up the Product Terms

Sum up all the product terms to find the final solution. So it will be: 25x^2 + 10x - 5x - 2. After simplifying this, we get 25x^2 + 5x - 2.

Key concepts

Distributive Property

Understanding the distributive property is a cornerstone of algebra. It's a rule that allows us to multiply a single term by each term in a parenthesis. In simpler terms, if you have an expression like \( a(b + c) \), the distributive property lets you 'distribute' the \( a \) across the \( b + c \), giving \( ab + ac \).

For example, in the exercise \((5x - 1)(5x + 2)\), we use the distributive property twice. First, we distribute \(5x\) over \((5x + 2)\), and then distribute \(-1\) over the same binomial. This property ensures that each element in the binomials is accounted for when calculating the product.

Polynomial Products

Multiplying polynomials, such as binomials in our exercise, involves finding the product of algebraic expressions with two or more terms. When we work with these polynomial products, we're essentially combining several applications of the distributive property.

To handle this systematically, we multiply each term in the first polynomial with every term in the second polynomial. In direct relation to our exercise, \((5x - 1)(5x + 2)\) turns into the four distinct multiplications as seen in the solution steps: \(5x\) times \(5x\), \(5x\) times \(2\), \(-1\) times \(5x\), and \(-1\) times \(2\). All of these products are then added together to find the final polynomial product.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (such as \(x\)), and operation symbols. It’s a vital concept in algebra because it represents quantities that may vary or are unknown, making it essential to comprehend how to manipulate these expressions.

In the context of our exercise, \((5x - 1)(5x + 2)\) is an example of a binomial multiplication where both binomials are algebraic expressions. The process of multiplying these expressions is more than calculating numbers; it involves understanding how to apply algebraic rules, such as the distributive property and combining like terms, to simplify and solve the expression.