Problem 29 - Algebra for College Students

Question

Find each product. $$(4 a b)\left(-2 a^{2} b\right)(7 a)$$

Step-by-Step Solution

Answer

-56a^4b^2

Step 1: Identify the Terms

The expression provided is \( (4ab)(-2a^{2}b)(7a) \). Here, we need to find the product of these three terms. Each term is a product of coefficients and variables: \(4ab\), \(-2a^2b\), and \(7a\).

Step 2: Multiply the Coefficients

First, multiply just the numerical coefficients: \(4\), \(-2\), and \(7\). \[4 \times (-2) \times 7 = -56\]

Step 3: Multiply the Variables

Now, multiply the variables. - For \(a\): \[ a imes a^2 imes a = a^{1+2+1} = a^4 \] - For \(b\): \[ b imes b = b^2 \]

Step 4: Combine the Results

Combine the results of the coefficients and the variables: Combine \(-56\) with \(a^4\) and \(b^2\), resulting in \[-56a^4b^2\]

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying expressions that contain multiple algebraic terms. In this context, you multiply each term by all other terms in different expressions. This process helps to simplify expressions and combine like terms. Consider polynomials with multiple components, such as variables with exponents and coefficients that need to be carefully distributed across other terms.

Consider the given polynomial expression:

• Terms: The product involves terms with different coefficients and variables, like \(4ab\), \(-2a^2b\), and \(7a\).
• Coefficients: Multiply the coefficients of each term together at the beginning. In this instance, the numbers \(4\), \(-2\), and \(7\) are multiplied to get \(-56\).
• Variables: After coefficients, combine the variables, ensuring to acknowledge their respective exponents during multiplication.

This method streamlines large expressions and reduces them into a single simplified term, making calculations easier in further algebraic operations.

Algebraic Variables

Algebraic variables are symbols representing numbers in expressions or equations. Often, they are denoted by letters like \(a\), \(b\), \(x\) or \(y\). These variables are essential because they allow for generalization and expression of mathematical ideas in a flexible way. In polynomial multiplication:

For instance, consider the variables in the expression \((4ab)(-2a^2b)(7a)\):

• \(a\) and \(b\) are variables that reflect terms across sections of a polynomial.
• Variables hold placeholders for numbers, presenting possibilities for diverse calculations.
• Through multiplication, these variables allow integration or simplification by following their coefficients and applied operations.

In the multiplication process, respecting the rules for variables, especially when linked to their coefficients and exponents, ensures an accurate and meaningful result.

Exponents in Algebra

Exponents in algebra represent how many times a number, or variable, is multiplied by itself. They are crucial in polynomials because they dictate the form and behavior of expressions. Understanding exponents properly allows for efficient algebraic problem-solving.

Examining the expression \((4ab)(-2a^2b)(7a)\) illuminates several principles:

• The base \(a\) appears with different exponents in each term. The exponent indicates repeated multiplication: \(a^2\) means \(a \times a\).
• While multiplying variables with exponents, you add the exponents if the bases are the same. For example, \(a^{1+2+1} = a^4\).
• Applying these operations to \(b\), which appears without an exponent initially, is simplified as \(b^1\). Combine exponents: \(b^1 \times b = b^2\).

Understanding and applying these exponent rules transform complex polynomial expressions into simpler, more approachable forms.