Problem 48 - Algebra 1: Concepts and Skills

Question

Use a vertical format to find the product. $$ (x+2)\left(x^{2}+3 x+5\right) $$

Step-by-Step Solution

Answer

The product of the two polynomials is \(x^{3} + 5x^{2} + 11x +10\).

Step 1: Distribute \(x\)

First, distribute \(x\) with \(x^{2}+3x+5\) which results in \(x*(x^{2}) + x*(3x) + x*(5) = x^{3}+3x^{2}+5x\)

Step 2: Distribute \(2\)

Next, distribute the \(2\) with \(x^{2}+3x+5\), resulting in \(2*(x^{2}) + 2*(3x) + 2*(5) = 2x^{2} +6x +10\)

Step 3: Combine like terms

Finally, combine the like terms from both expressions: \(x^{3}+3x^{2}+5x+2x^{2}+6x+10 = x^{3}+5x^{2}+11x+10\)

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions and solve equations. It involves multiplying each term inside a bracket by a term outside the bracket. This is especially useful when dealing with polynomial multiplication. The property is expressed as:\[ a(b + c) = ab + ac \]

• In this property, the "a" term outside the bracket is distributed to each term inside the parentheses.
• Each term inside the bracket gets multiplied by the term outside.

Using the distributive property in polynomial multiplication like \[(x+2)(x^{2}+3x+5)\] means applying it twice: once with the first term "x" and once with the second term "2". This gives us two separate multiplication operations resulting in new expressions.
This property makes it straightforward to handle operations without missing any components. By breaking it down like this, math becomes simpler and more systematic.

Combining Like Terms

Combining like terms is a critical skill in simplifying algebraic expressions and polynomials. Like terms are those that contain the same variable raised to the same power. This means their coefficients can be added or subtracted while keeping the variable part unchanged.Consider the product we obtained:\[x^{3}+3x^{2}+5x + 2x^{2}+6x +10\]

• We look for terms that have identical variable parts.
• In the example, \(3x^{2}\) and \(2x^{2}\) are like terms.
• Similarly, \(5x\) and \(6x\) are like terms.

By combining these, we add their coefficients:

• \(3x^{2} + 2x^{2} = 5x^{2}\)
• \(5x + 6x = 11x\)

This results in the simpler expression \(x^{3} + 5x^{2} + 11x + 10\), allowing for a clear and organized understanding of the polynomial. This step is crucial as it consolidates our expression into its simplest form.

Algebraic Expressions

Algebraic expressions are at the core of algebra. They are combinations of variables, numbers, and operations (like addition and multiplication). Unlike equations, they do not have an equality sign, and they represent values rather than specific quantities.When dealing with algebraic expressions such as \[(x+2)(x^{2}+3x+5),\] we manipulate them using operations like polynomial multiplication. This involves techniques such as the distributive property and combining like terms.Algebraic expressions can be simple, containing just one term, or more complex with many terms. A good grasp of expressions is essential as they form the foundation for writing and solving equations. When working with expressions:

• Identify the terms involved.
• Use the distributive property to manage multiplication.
• Combine like terms to simplify the expression.

Understanding and simplifying algebraic expressions helps in tackling more advanced algebraic problems and lays the groundwork for calculus and beyond.