Problem 36 - Algebra and Trigonometry

Question

Find the sum, difference, or product. 3\(x^{3}\left(x^{4}-4 x^{2}+5\right)\)

Step-by-Step Solution

Answer

The expanded expression is \(3x^{7} - 12x^{5} + 15x^{3}\).

Step 1: Expand the expression

We start by distributing the term outside the parentheses to each term inside the parentheses. The expression given is \(3x^{3}(x^{4} - 4x^{2} + 5)\).

Step 2: Distribute the first term

Multiply \(3x^{3}\) by \(x^{4}\). The result is \(3x^{3} \times x^{4} = 3x^{7}\).

Step 3: Distribute the second term

Multiply \(3x^{3}\) by \(-4x^{2}\). The result is \(3x^{3} \times -4x^{2} = -12x^{5}\).

Step 4: Distribute the third term

Multiply \(3x^{3}\) by \(5\). The result is \(3x^{3} \times 5 = 15x^{3}\).

Step 5: Combine all terms

Combine all the distributed terms to form the expanded expression: \(3x^{7} - 12x^{5} + 15x^{3}\).

Key Concepts

Multiplication of Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured with terms like powers and constants. When we talk about multiplying polynomials, we are looking at taking each term from one polynomial and multiplying it by every term in another polynomial.

This might sound complicated, but the concept follows simple arithmetic rules. One of the most basic multiplication operations within polynomials is the Distributive Property, which we will discuss in the next section. In the given exercise, a polynomial was multiplied by a monomial (a single-term polynomial). When multiplying, follow these easy steps:

• Identify each term of the polynomial to be expanded.
• Multiply the external term (in our case, the monomial) by each term inside the polynomial.
• Combine and simplify the results by summing similar powers or terms.

It's like the familiar activity of distributing cards but applied to algebraic expressions! You "give out" the external term to each participant (term) inside the parentheses.

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows you to multiply a single term across two or more terms inside a set of parentheses. Essentially, it's the bridge that allows multiplication to "distribute" over addition and subtraction inside an expression. For example, in the exercise, we used the Distributive Property to multiply the term outside, 3\(x^3\), to each term inside the parentheses: \(x^4, -4x^2,\) and \(5\).

Here's how the Distributive Property works with polynomials:

• Take the term outside the parenthesis and multiply it by each term inside.
• Remember to perform the multiplication individually, respecting both the coefficients and the powers of the variables.
• Sums or differences inside the brackets maintain their operation signs, while the multiplication is completed.

Mathematically, it's presented as \(a(b + c) = ab + ac\). In our case, the property helped simplify our polynomial multiplication resulting in terms like \(3x^7\), \(-12x^5\), and \(15x^3\).

Algebraic Expressions

Algebraic expressions are the building blocks of algebra. They consist of numbers and variables connected through arithmetic operations like addition, subtraction, multiplication, and division. Understanding them is crucial, as they are more than just strings of numbers and letters.

To break it down:

• Variables: Symbols that stand in for unknown values, like \(x\) in our exercise.
• Coefficients: Numbers that multiply the variables, such as the 3 in \(3x^3\).
• Terms: Compositions of variables and coefficients, like \(3x^3\), that are added or subtracted from each other within an expression.

Algebraic expressions don't always have to resolve to a single number; instead, they represent a range of possibilities and can be simplified or manipulated in multiple ways, such as through factoring or expanding. In the context of polynomial multiplication, understanding each part of the expression allows for clearer insight into the operations being performed and the resulting mathematical relationships.