Question

In Exercises 17–24, fi nd the product. \(7 x^3\left(5 x^2+3 x+1\right)\)

Short answer

The product is \(35x^5 + 21x^4 + 7x^3\).

Step-by-step solution

Identify the terms in the polynomial

In the given polynomial \(5x^2+3x+1\), there are three terms: \(5x^2\), \(3x\), and \(1\).

Multiply the single term with each term of the polynomial

Distribute \(7x^3\) across the terms of the trinomial: \(7x^3 \times 5x^2 = 35x^5\), \(7x^3 \times 3x = 21x^4\), \(7x^3 \times 1 = 7x^3\).

Write the final expression

Combine all the results from step 2 to write the final expression: \(35x^5+21x^4+7x^3\).

Key concepts

Understanding the Distributive Property

The distributive property is a key concept in algebra that allows us to simplify expressions when multiplying polynomials. It involves distributing a single term into a polynomial by multiplying it with each term within the parenthesis. This process ensures each term is accounted for, maintaining the integrity of the algebraic expression.
When we apply the distributive property to a polynomial multiplication problem like \(7x^3(5x^2+3x+1)\), we:

• Multiply each term inside the bracket by the term outside, which in this case is \(7x^3\).
• Break down the complex multiplication process into manageable steps.
• Simplify each resulting term, paying attention to the rules of exponents, such as adding the exponents when multiplying like bases.

The distributive property helps us keep track of all variables and coefficients systematically and is essential for solving algebraic equations effectively.

Exploring Trinomials

A trinomial is a type of polynomial that contains exactly three terms. In the example \(5x^2 + 3x + 1\), each variation in the power of \(x\) represents a distinct term.
Understanding these components is crucial when multiplying expressions. Each term includes a coefficient (a constant that multiplies the variable), and its variable raised to a specific power.
Characteristics of trinomials include:

• They may contain variables raised to different powers, like \(x^2\) in one term and \(x\) in another.
• Trinomials are common in quadratic equations, though they are not limited to them.
• The coefficients provide a multiplier for the variable part, affecting the width or steepness in graphs.

Understanding how these individual terms interact during multiplication is key to leveraging the distributive property effectively.

Decoding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. In algebra, these expressions are used to represent relationships, describe real-world phenomena, or solve problems.
An algebraic expression like \(7x^3(5x^2+3x+1)\) consists of:

• Variables, usually represented by letters like \(x\), which stand for unknown or changeable values.
• Coefficients, which are constants working with variables to create expressions.
• Operations such as addition, subtraction, and multiplication that connect terms.
• Terms, which can be a standalone variable, a number, or a combination of both, separated by operators.

Algebraic expressions form the foundation of equations and functions and require careful handling when evaluated or simplified. Mastering these concepts aids in decoding complex formulas and solving a wide range of mathematical problems.