Problem 71 - Algebra and Trigonometry

Question

Perform the indicated operations, and simplify. \((x+2)\left(x^{2}+2 x+3\right)\)

Step-by-Step Solution

Answer

The result is \(x^3 + 4x^2 + 7x + 6\).

Step 1: Distribute the First Term

Start by distributing the first term in the binomial, which is \(x\). Multiply \(x\) with each term inside the trinomial \((x^2 + 2x + 3)\). This results in: \[ x imes x^2 + x imes 2x + x imes 3. \] Which simplifies to:\[ x^3 + 2x^2 + 3x. \]

Step 2: Distribute the Second Term

Next, distribute the second term in the binomial, which is \(2\). Multiply \(2\) with each term inside the trinomial \((x^2 + 2x + 3)\). The result of this distribution is:\[ 2 imes x^2 + 2 imes 2x + 2 imes 3, \]which simplifies to:\[ 2x^2 + 4x + 6. \]

Step 3: Combine Like Terms

Now, add the results from Step 1 and Step 2:\[ x^3 + 2x^2 + 3x + 2x^2 + 4x + 6.\]Combine like terms to simplify further. Group the \(x^2\) terms and the \(x\) terms:\[ x^3 + (2x^2 + 2x^2) + (3x + 4x) + 6. \]This combines to:\[ x^3 + 4x^2 + 7x + 6. \]

Key Concepts

Distributive Property

The distributive property is an essential tool in algebra that allows us to multiply a single term by each term within a group, such as a binomial or trinomial. This property is often phrased as "multiply outside term with each inside term". For instance, in the expression \(a(b+c)\), using the distributive property leads to \(ab + ac\).
This property ensures that we account for the influence of every term during multiplication, and it helps break down complex problems into more manageable parts. When faced with a multiplication involving a polynomial, just remember to apply this property consistently to each term in the sequence.
In the exercise, we multiply \(x+2\) with \(x^2+2x+3\). We distribute both \(x\) and \(2\) over every term in the trinomial, effectively applying the distributive property twice.

Binomial

A binomial is a polynomial that contains exactly two terms. These terms can be numbers, variables, or the product of numbers and variables, added or subtracted from one another. For example, \(x + 2\) is a binomial because it consists of two distinct, non-zero terms held together by a plus sign.
Binomials are foundational in algebra, introducing us to the operations of addition and multiplication within polynomials.

• They can be simple, like \(x + 1\), or more complex, like \(-3a + 4b\).
• Operations involving binomials often include combining them with other polynomials, or multiplying them, as seen in the exercise.

Approaching binomials with an understanding of the distributive property makes operations smoother and prepares you for more complex algebraic expressions, like trinomials or polynomials with multiple terms.

Trinomial

A trinomial is a type of polynomial consisting of three distinct terms. It is an extension of the idea presented by binomials, adding another layer of complexity and therefore more terms to consider during algebraic operations. For example, \(x^2 + 2x + 3\) is a trinomial because it includes three separate components each with its factor.
Trinomials allow for the practice of distributing over more terms, as seen in the current multiplication exercise:

• The first term \(x\) is distributed across each term in \((x^2 + 2x + 3)\).
• Then, the second term \(2\) is distributed in the same manner.

In operations involving trinomials, each term must be considered individually during multiplication or addition processes, making the accurate application of the distributive property crucial.

Combining Like Terms

Combining like terms is the process of simplifying algebraic expressions by merging terms that have the same variable and power. It's like tidying up a room by grouping similar items together. When you have terms like \(2x\) and \(3x\), you can combine them into \(5x\), as they are 'like terms' both involving \(x\).
In our exercise, after distributing and obtaining new terms from both parents of the binomial, we encounter like terms with \(x^2\) and \(x\) elements.

• Combine \(2x^2\) and \(2x^2\) to get \(4x^2\).
• Combine \(3x\) and \(4x\) to form \(7x\).

This step is critical for simplifying expressions to their most compact and easily understandable form. Always look out for like terms after performing steps like distribution to ensure your final equation is as clear and concise as possible.