Problem 25 - Algebra and Trigonometry

Question

25–30 ? Factor the expression by grouping terms. $$ x^{3}+4 x^{2}+x+4 $$

Step-by-Step Solution

Answer

The expression factors to \((x + 4)(x^2 + 1)\).

Step 1: Identify Groupable Terms

First, we need to group the terms in pairs that make it possible to factor by grouping. We will divide the expression into two groups: \[ (x^3 + 4x^2) + (x + 4) \]

Step 2: Factor Out the Greatest Common Factor from Each Group

Look at each group separately and factor out the greatest common factor.- From the first group \((x^3 + 4x^2)\), factor out \(x^2\): \[ x^2(x + 4) \]- From the second group \((x + 4)\), there is no need to factor as it is already simple: \[ 1(x + 4) \]

Step 3: Combine Factored Terms

Now that both groups have a common factor \((x + 4)\), we can factor this common term out of the entire expression:\[ (x + 4)(x^2 + 1) \]

Step 4: Check the Factored Expression

To ensure accuracy, distribute the factored terms to verify that they match the original expression:- Distribute \((x + 4)\) into \((x^2 + 1)\): - \((x)(x^2) = x^3\) - \((x)(1) = x\) - \((4)(x^2) = 4x^2\) - \((4)(1) = 4\)Combine these terms: \(x^3 + 4x^2 + x + 4\), confirming the factorization is correct.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is an important concept in algebra that helps simplify expressions by finding the largest factor shared by all terms in an expression. When you work with polynomials, the GCF allows you to break down complex expressions into simpler terms.
In the case of our expression, we first group the terms into pairs as:

• \((x^3 + 4x^2)\)
• \((x + 4)\)

By factoring out the GCF from each group, you simplify the expression. For the first pair \((x^3 + 4x^2)\), the GCF is \(x^2\), simplifying it to \(x^2(x + 4)\).
Finding and using the GCF is a foundational step in factoring by grouping and greatly aids in making expressions more manageable.

Polynomial Factorization

Polynomial Factorization involves breaking down a polynomial into a product of simpler polynomials. This is often used for simplifying expressions or solving equations. Factorization helps by expressing the polynomial in terms of its factors, which are polynomials of lower degree or constants.
To factor the given polynomial \(x^3 + 4x^2 + x + 4\), start by recognizing common terms. Group them strategically:

• \((x^3 + 4x^2)\)
• \((x + 4)\)

The goal is to simplify each group individually using their GCFs, then combine them to express the entire polynomial as a product of factors.
For example, from the expression \((x^3 + 4x^2) + (x + 4)\), the factored form is \((x + 4)(x^2 + 1)\). This method is fundamental in simplifying complex algebraic expressions and preparing for solving polynomial equations.

Algebraic Expression

An Algebraic Expression is a combination of variables, numbers, and operations. Understanding the components of algebraic expressions is crucial for their manipulation and simplification. They are used to represent mathematical concepts symbolically.
In our exercise, the expression \(x^3 + 4x^2 + x + 4\) is an algebraic expression consisting of terms with variables and coefficients:

• \(x^3\): A variable raised to the power of 3.
• \(4x^2\): A term with a coefficient of 4 and a variable raised to the power of 2.
• \(x\): A single variable term.
• \(4\): A constant term.

Understanding the structure of an algebraic expression helps in techniques like factoring by grouping, which involves rearranging terms for simplification. Mastery of algebraic expressions is essential in algebra and beyond, facilitating problem-solving and analysis in a broad range of mathematical scenarios.