Problem 52 - Algebra 1: Concepts and Skills

Question

Use a horizontal format to find the product. $$ (x+4)\left(x^{2}-2 x+3\right) $$

Step-by-Step Solution

Answer

The product of \( (x+4) \) and \( \left(x^{2}-2 x+3\right) \) is \( x^{3}+2x^{2}-5x+12 \).

Step 1: Distribute terms in the first polynomial into the second

The first step is to distribute \( x+4 \) to \( x^{2}-2 x+3 \). Here, the term \( x \) must be multiplied by every term in \( x^{2}-2 x+3 \), producing \( x*x^{2} \), \( x*(-2x) \), and \( x*3 \). Next, \( 4 \) will also be multiplied by each term in \( x^{2}-2 x+3 \), producing \( 4*x^{2} \), \( 4*(-2x) \), and \( 4*3 \).

Step 2: Simplify each product

Perform each multiplication operation calculated in Step 1. This gives us \( x^{3} \), \( -2x^{2} \), \( 3x \), \( 4x^{2} \), \( -8x \), and \( 12 \).

Step 3: Combine like terms

Add together any terms in the polynomial that have the same degree. We notice that there are two terms of degree 2 and two terms of degree 1. Combining these gives us \( x^{3}+2x^{2}+-5x+12 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve equations. It involves distributing one term across others within parentheses. This property states that for any numbers or variables \( a \), \( b \), and \( c \), the expression \( a(b + c) \) is equivalent to \( ab + ac \).

In our problem, we have \((x+4)\) multiplied by \((x^2-2x+3)\). We apply the distributive property by multiplying each term in \((x+4)\) with every term in \((x^2-2x+3)\).

• First, multiply \(x\) by each term: \(x \cdot x^2\), \(x \cdot (-2x)\), and \(x \cdot 3\).
• Next, multiply \(4\) by each term: \(4 \cdot x^2\), \(4 \cdot (-2x)\), and \(4 \cdot 3\).

This manipulation allows us to break down the expression into manageable pieces, facilitating the multiplication process.

Combining Like Terms

Combining like terms is an important step in simplifying polynomials. It involves adding or subtracting terms that have the same variable raised to the same power. Like terms have identical variable parts. For instance, \(2x^2\) and \(-2x^2\) are like terms because they both contain \(x^2\).

After using the distributive property in our exercise, we end up with these terms: \(x^3\), \(-2x^2\), \(3x\), \(4x^2\), \(-8x\), and \(12\). The goal is to combine terms that have the same powers:

• Combine \(-2x^2\) and \(4x^2\) to get \(2x^2\).
• Combine \(3x\) and \(-8x\) to get \(-5x\).

Doing this helps to simplify the expression to \(x^3 + 2x^2 - 5x + 12\), giving us a cleaner and more workable polynomial.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent mathematical relationships and patterns. An algebraic expression might include terms like \(3x\), \(2x^2\), or constants like \(12\). In algebra, variables such as \(x\) can represent unknown quantities.

In our exercise, \((x+4)(x^2-2x+3)\) is an algebraic expression composed of two polynomials. To work with these expressions, it's necessary to perform operations such as multiplication and combining like terms.

• Understand variable terms: \(x^3\), \(x^2\), \(x\).
• Identify constants: numbers without variables like \(12\).

Learning to manipulate algebraic expressions is essential for solving equations and understanding mathematical concepts.