Problem 29 - Elementary and Intermediate Algebra

Question

Multiply. See Example 2. $$ -2 x^{3}\left(3 x^{2}-x+1\right) $$

Step-by-Step Solution

Answer

The result is \(-6x^5 + 2x^4 - 2x^3\).

Step 1: Distribute the First Term

First, distribute the term \( -2x^3 \) to the first term inside the parentheses \( 3x^2 \). We do this by multiplying them together: \[ -2x^3 imes 3x^2 = -6x^{3+2} = -6x^5. \] This gives us the first term of the expanded polynomial.

Step 2: Distribute the Second Term

Next, distribute \( -2x^3 \) to the second term inside the parentheses \( -x \). Multiply them together: \[ -2x^3 imes (-x) = 2x^{3+1} = 2x^4. \] This becomes the second term of the expanded polynomial.

Step 3: Distribute the Third Term

Finally, distribute \( -2x^3 \) to the last term \( 1 \). By multiplying: \[ -2x^3 imes 1 = -2x^3. \] This term completes the expansion.

Step 4: Combine the Products

Combine the results of all the distributed products to form the final expression: \[-6x^5 + 2x^4 - 2x^3. \] This is the polynomial obtained after multiplying through.

Key Concepts

Distributive Property

The Distributive Property is a key arithmetic concept that allows us to simplify and solve many math problems. It helps us understand how to multiply a number or expression by a sum or difference within parentheses. When using the distributive property, each term inside the parentheses is multiplied by the term outside the parentheses. This method ensures that every part of the expression is accounted for and helps break down complex calculations into simpler steps.

• "Distribute" a term across the terms in parentheses.
• Multiply the term outside by each term inside, one by one.
• Combine the results to form the expanded expression.

For example, in the exercise, the term \(-2x^3\) is distributed across the polynomial \(3x^2 - x + 1\). This creates individual multiplication steps that can then be solved separately, making the overall expression easy to handle and expand.

Monomials

A monomial is a simple algebraic expression composed of a single term. This means it could be a constant, a variable, or variables multiplied together, perhaps with a coefficient in front. Monomials are the building blocks of more complex polynomial expressions.

• Single, non-zero term algebraic expressions.
• Can be numbers, variables, or a combination of both.
• Represented as \(a x^n\), where \(a\) is a coefficient and \(n\) is an exponent.

In the exercise, \(-2x^3\) is a monomial. It features a coefficient of -2 and a variable \(x\) raised to the power of 3 (exponent). Multiplying a monomial by other terms follows basic rules of arithmetic, which include using the distributive property to ensure that each part of the expression is accounted for.

Exponents

Exponents are an important component when dealing with algebraic expressions, particularly monomials. An exponent indicates how many times a base number is multiplied by itself. This concept is crucial when expanding expressions and solving problems involving power of numbers.

• An exponent shows repeated multiplication of a number by itself.
• Written as a small number above and to the right of the base, e.g., \(x^2\).
• Follow specific rules, like \(a^m \times a^n = a^{m+n}\).

In the exercise, exponents help in simplifying expressions like calculating \(-2x^3 \times 3x^2 = -6x^5\), following the rule of adding exponents when multiplying terms with the same base. Understanding these properties is essential for managing more complicated expressions.

Polynomial Expression

Polynomial expressions are algebraic expressions that consist of variables, coefficients, and exponents. These expressions can have multiple terms, which could include any combination of monomials. Each term in a polynomial has its own coefficient and may include variables raised to non-negative integer exponents.

• Polynomials consist of terms added or subtracted together.
• Each term’s format is usually \(ax^n\), where a is any real number, and n is a non-negative integer.
• Can range from simple expressions like \(x^2 + 1\) to more complex forms with several terms.

In this exercise, the expression \(-2x^3(3x^2-x+1)\) represents a multiplication of a monomial with a polynomial. This process involves expanding the polynomial using the distributive property, handling each part distinctly to form the complete expanded polynomial: \(-6x^5 + 2x^4 - 2x^3\). Understanding polynomial expressions and how to manipulate them with operations such as multiplication is key to unlocking more advanced algebraic concepts.