Problem 32 - Algebra for College Students

Question

Find each product. $$\left(-3 a^{2} b\right)\left(-a b^{2}\right)(-7 a)$$

Step-by-Step Solution

Answer

The product is \(21a^4b^3\).

Step 1: Multiply the coefficients

First, multiply the numerical coefficients from each term. Here, the coefficients are -3, -1 (implied in \(-ab^2\)), and -7. Combining these gives \((-3) imes (-1) imes (-7) = 21\).

Step 2: Combine the powers of 'a'

Next, multiply the powers of \(a\). Add the exponents together: \(a^2 imes a^1 imes a^1 = a^{2+1+1} = a^4\).

Step 3: Combine the powers of 'b'

Similarly, multiply the powers of \(b\). Add the exponents: \(b^1 imes b^2 = b^{1+2} = b^3\).

Step 4: Write the final expression

Combine the results from Step 1, Step 2, and Step 3. The expression becomes \(21a^4b^3\).

Key Concepts

Coefficients

In algebra, coefficients are the numbers in front of variables that indicate how many times a term is multiplied. They are crucial in polynomial multiplication. These coefficients can be positive or negative numbers. In our example, the first coefficient is -3 in (-3a^2b), -1 in (-ab^2), and -7 in (-7a).

• We treat them as regular numbers and multiply them as such.
• Negative signs follow the rules of multiplication: a negative times a negative gives a positive.

First, multiply -3 by -1, resulting in 3, and then multiply 3 by -7, resulting in 21. These steps clarify how to simplify the multiplication of coefficients.

Exponents

Exponents describe how often a number, termed the base, is multiplied by itself. In polynomial multiplication, we often deal with variables raised to a power. Combining terms often requires adding exponents. For example, consider multiplying the powers of:

• \(a^2 \times a^1 \times a^1\)
• This simplifies to \(a^{2+1+1}\), which results in \(a^4\).

• Another example involves \(b\), where powers are combined as per: \(b^1 \times b^2 = b^{1+2} = b^3\).

Remember, we use this process only when multiplying like bases.

Monomials

Monomials are the simplest form of algebraic expressions, consisting of a single term. Each monomial is made up of three parts: a coefficient, a variable, and an exponent. In our case, each term like \((-3a^2b)\) is a monomial.

• They are individual components in polynomial expressions.
• Monomials can be identified by a single term structure such as \(5x^3y\).

They are ideal for multiplication since each factor is standalone. Reduce complexity by handling monomials independently, before combining them back into the algebraic structure.

Algebraic Expressions

Algebraic expressions are mathematical phrases that involve variables and operations. They combine numbers, coefficients, variables, and arithmetic operations like addition, subtraction, multiplication, and division. An algebraic expression can be as simple as a monomial or more complex like polynomials.

• Each term within an expression might differ, featuring varied coefficients and exponents.
• Although they look complicated, breaking them down into smaller parts (monomials) aids in understanding.
• The exercise demonstrated is a product of three monomials, forming a comprehensive algebraic expression.

By decomposing the expression into smaller segments, it provides clarity and simplifies solving, especially in polynomial multiplication.