Problem 54 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 4 k\left(3 k^{2}+2 k+6\right)+k\left(5 k^{2}+k\right)+16 $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \(4k(3k^2 + 2k + 6)+ (k^3 + k^2 + 16)\). Answer: \(17k^3 + 9k^2 + 24k + 16\)

Step 1: Distribute terms

Multiply the terms outside the parentheses by the corresponding terms inside the parentheses. For both terms, multiply the first term (4k and k) by the terms in the parentheses. $$ 12 k^{3}+8 k^{2}+24 k+5 k^{3}+k^{2}+16 $$

Step 2: Combine like terms

Now, combine the terms that have the same variable and degree (power). In this case, we have three different types of terms: \(k^{3}\), \(k^{2}\), and \(k\). $$ (12k^3 + 5k^3) + (8k^2 + k^2) + 24k + 16 $$

Step 3: Simplify the expression

Add the coefficients of the like terms to simplify the expression. $$ 17 k^3 + 9 k^2 + 24 k + 16 $$ The simplified algebraic expression is \(17k^3 + 9k^2 + 24k + 16\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Think of them like sentences in the language of algebra. They are structured using two main components: coefficients, which are the numerical parts, and variables, which are often represented by letters like \( k \) or \( x \). Each variable can have an exponent, indicating how many times it's multiplied by itself. The expression \( 4k(3k^2 + 2k + 6) \) contains several terms, like \( 3k^2 \) and \( 2k \), combined by addition or subtraction.
These expressions can grow in complexity, involving multiple operations and parentheses, but the rules to manipulate them remain constant. Learning to work with algebraic expressions builds a foundational understanding for all the other operations and rules in algebra that you'll encounter.

Distributive Property

The distributive property is a fundamental rule in algebra that helps to break down expressions into simpler components. This property states that \( a(b + c) = ab + ac \). In simple terms, you multiply the term outside the parentheses by each term inside.
Consider the expression \( 4k(3k^2 + 2k + 6) \). Here, the 4k is distributed across each term inside the parentheses, resulting in \( 12k^3 + 8k^2 + 24k \). Working through these multiplications correctly is crucial, as it sets the stage for combining like terms later. The distributive property is about unraveling, making complex expressions more manageable.

Combining Like Terms

Once you've applied the distributive property, you'll often find yourself with several terms to sort through. Combining like terms is the process of simplifying those terms with the same variables and exponents. Consider \( 12k^3 \) and \( 5k^3 \) in our expression; both have the same variable part and exponent, making them like terms.
To combine them, simply add their coefficients: \( 12k^3 + 5k^3 = 17k^3 \). The same goes for \( 8k^2 \) and \( k^2 \), which combine to form \( 9k^2 \). By neatly organizing and combining like terms, the expression becomes clearer and easier to work with.

Polynomials

The term polynomial refers to an algebraic expression that consists of multiple terms. These can be constants, variables, or a combination of both, each term separated by addition or subtraction. Our expression, \( 17k^3 + 9k^2 + 24k + 16 \), is a classic example.

• It is a polynomial of the third degree because the highest power of the variable \( k \) is 3.
• Each separate part such as \( 17k^3 \), \( 9k^2 \), \( 24k \), and the whole number 16 are called terms.

Polynomials are versatile and appear frequently in mathematical equations and real-world applications. They set the stage for more advanced mathematical concepts, such as factoring and solving equations.