Question

Multiply. $$-2 x^{3}(5 x-8)$$

Short answer

Question: Multiply the monomial -2x^3 with the binomial (5x - 8). Answer: -10x^4 + 16x^3

Step-by-step solution

Identify the monomial and binomial

Here, the monomial is \(-2x^3\) and the binomial is \((5x - 8)\).

Apply the distributive property

Now, we will multiply the monomial with each term in the binomial: $$-2x^3(5x) + (-2x^3)(-8)$$

Multiply the terms

Perform the multiplications: $$-10x^4 + 16x^3$$

Write the final result

Our final answer after multiplying the monomial and binomial is: $$-10x^4 + 16x^3$$

Key concepts

Monomial

A monomial is a single term algebraic expression composed of variables, coefficients, and constants. It can include any combination of these three components, but there must be no addition or subtraction involved within a single monomial. A monomial has only one term. For example, in the given exercise, \(-2x^3\) is a monomial.Key characteristics of a monomial include:

• Single term without addition or subtraction.
• Consists only of multiplication or division.
• Can include numbers and one or more variables.
• Power of each variable is a whole number.

In our exercise, the monomial \(-2x^3\) includes a coefficient, which is \-2\, and a variable part, \x^3\, representing that \x\ is raised to the power of 3.

Binomial

A binomial is an algebraic expression that contains exactly two terms. These two terms are separated by either a plus \(+\) or minus \(-\) sign. In the exercise, \(5x - 8\) is identified as a binomial.Some important properties of a binomial are:

• Consists of two distinct terms.
• Separated by addition or subtraction.
• Each term can have its own coefficient and variables.

In the example provided, the binomial has two terms: \5x\ and \-8\. Here, \5x\ is a term with a coefficient of 5 and a variable of \x\, while \-8\ is a constant term.

Multiplication of Polynomials

When multiplying polynomials, the distributive property is a powerful tool. This property enables us to multiply a single term with multiple terms inside parentheses. In other words, we distribute the monomial to each term of the polynomial separately. This technique simplifies the process of multiplying different polynomial expressions.Let's break down the multiplication of a monomial \(-2x^3\) with a binomial \(5x - 8\):

• First, multiply the monomial \(-2x^3\) by the first term of the binomial \(5x\). This yields \(-2x^3 \times 5x = -10x^4\).
• Next, multiply the monomial by the second term \(-8\) to get \(-2x^3 \times (-8) = 16x^3\).

By following these steps, we combine all distributed products to arrive at the final result. In this example, the final result is \-10x^4 + 16x^3\, showcasing the comprehensive multiplication of a monomial with a binomial.