Question

Multiply and simplify. $$(x+3.88)\left(x^{3}-2.15 x-6.03\right)$$

Short answer

\(x^4 + 3.88x^3 - 2.15x^2 - 14.13x - 23.4144\)

Step-by-step solution

Distributive Property

Apply the distributive property (also known as the FOIL method for binomials) to multiply each term in the first expression by each term in the second.

Multiply the Terms

Multiply the term outside the parentheses, x, by each of the three terms inside the parentheses. Then multiply 3.88 by each term inside the parentheses.

Combine Like Terms

After multiplication, combine like terms to simplify the expression if necessary.

Key concepts

Distributive Property

Understanding the distributive property is essential when working with polynomial multiplication. This property is a cornerstone of basic algebra and allows us to multiply a single term by a polynomial, essentially 'distributing' the multiplication over each term within the brackets.

For instance, when you have an expression like \((x + a)(b + cx + d)\), you break it down by multiplying \(x\) by every term in the second polynomial and then do the same with \(a\). The distributive property ensures that every term in the first polynomial interacts with every term in the second.

Let's apply this to the given exercise \((x+3.88)(x^3-2.15x-6.03)\). Here, we distribute the first term \(x\) across the three terms in the second polynomial, then distribute \(3.88\) across those same three terms. It is this step-by-step distribution that leads us into multiplying terms individually and lays down the foundation for combining like terms later on.

FOIL Method

The FOIL method is an acronym that stands for First, Outer, Inner, Last, and it is commonly used to multiply two binomials. However, the principle behind FOIL can be extended when multiplying any polynomials, not just binomials.

When dealing with more complex situations, such as in our exercise, you treat every term in the first polynomial as a 'First' and put it through the FOIL process with every term in the second polynomial. For instance, you begin by multiplying the 'First' terms in both polynomials, then the 'Outer' terms, and so on.

While the traditional FOIL method is not precisely applied in our exercise because the second polynomial has more than two terms, the underlying approach of structured multiplication is. Each term in the first polynomial (\(x\) and \(+3.88\)) is multiplied in turn by each term in the second polynomial (\(x^3\), \(-2.15x\), and \(-6.03\)), akin to a broader application of the FOIL method.

Combining Like Terms

After multiplying the polynomials using the distributive property or FOIL method, we often end up with multiple terms that can be combined to simplify the expression. Combining like terms is the process of adding or subtracting terms with the same variable raised to the same power.

Let's consider the partially expanded version of the given problem after employing the distributive property: \(x^4 - 2.15x^2 - 6.03x + 3.88x^3 - 8.346x - 23.4544\). Here we search for terms that are 'like' or share the same variables and exponents. In this case, we don't have any like terms, so combining them does not further simplify the expression.

However, it's crucial to always search for like terms after distributing and multiplying as they can often be found and simplified, which reduces the polynomial's complexity and makes it more manageable.