Problem 18 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. \((2 x+3)\left(x^{2}+x-5\right)\)

Step-by-Step Solution

Answer

The result is \(2x^3 + 5x^2 - 7x - 15\).

Step 1: Distribute the first term

To perform the operations, we start by distributing the first term of the binomial, which is \(2x\), across the trinomial \((x^2 + x - 5)\).Distributing \(2x\):- \(2x \cdot x^2 = 2x^3\)- \(2x \cdot x = 2x^2\)- \(2x \cdot (-5) = -10x\)

Step 2: Distribute the second term

Next, distribute the second term of the binomial, which is \(3\), across the trinomial \((x^2 + x - 5)\).Distributing \(3\):- \(3 \cdot x^2 = 3x^2\)- \(3 \cdot x = 3x\)- \(3 \cdot (-5) = -15\)

Step 3: Combine like terms

Now, add all the expressions from the distribution steps together and combine like terms:\[ 2x^3 + 2x^2 - 10x + 3x^2 + 3x - 15 \]- Combine \(2x^2\) and \(3x^2\) to get \(5x^2\)- Combine \(-10x\) and \(3x\) to get \(-7x\)Thus, the expression becomes:\[ 2x^3 + 5x^2 - 7x - 15 \]

Step 4: Write the result in simplest form

The expression \(2x^3 + 5x^2 - 7x - 15\) is already simplified, as there are no further like terms to combine.

Key Concepts

Binomial

In algebra, a binomial is a polynomial expression that contains exactly two terms. These terms are connected by either a plus "+" or minus "-" sign. Binomials are a fundamental concept because they are often used as a building block for more complex algebraic expressions.
In our exercise, the binomial is \(2x + 3\). This particular expression contains:

• A first term: \(2x\), where \(2\) is the constant coefficient and \(x\) is the variable.
• A second term: \(3\), which is a constant value or simply a number without a variable.

Understanding how binomials interact with other expressions, especially through multiplication, is essential for solving polynomial equations.

Trinomial

A trinomial is a type of polynomial containing exactly three distinct terms. These terms are also typically connected by "+" or "-" signs, similar to binomials. The concept of trinomials is central to polynomial multiplication when handling more complex equations.
In the given exercise, the trinomial is \(x^2 + x - 5\). It has:

• A first term: \(x^2\), which is a quadratic term, indicating the degree of the polynomial as 2 for this term.
• A middle term: \(x\), a linear term with a degree of 1.
• A last term: \(-5\), a constant term.

Trinomials are crucial in algebra because they can simplify the process of factorization and are integral to solving quadratic equations. When multiplied with a binomial, they require the use of the distributive property to simplify the expressions.

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to multiply a single term by each term inside a parenthesis separately and then sum the results. It has the mathematical form \(a(b + c) = ab + ac\). This property is essential in polynomial multiplication because it simplifies complex expressions.
In the exercise, we apply the distributive property in two main steps:

• First, distribute the first term of the binomial, \(2x\), across the entire trinomial \(x^2 + x - 5\).
• Next, distribute the second term of the binomial, \(3\), across the same trinomial.

These distribution steps break down the polynomial multiplication into smaller and more manageable calculations. Simplifying large expressions becomes more systematic with this property, making it invaluable in algebra.

Combining Like Terms

Combining like terms is a critical step in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variables raised to the same powers.
For instance, terms like \(2x^2\) and \(3x^2\) are like terms because both contain \(x^2\). Similarly, \(-10x\) and \(3x\) are like terms due to their common \(x\) variable.
In this exercise, after using the distributive property, we obtain several expressions. The goal is to combine these like terms:

• Add \(2x^2 + 3x^2\) to get \(5x^2\).
• Combine \(-10x\) and \(3x\) resulting in \(-7x\).

Combining like terms results in a streamlined expression, making it easier to read and manage, especially when inputting this simplified form as the final answer.