Problem 31 - Algebra for College Students

Question

Find each product. $$(-a b)(-3 a b)(-6 a b)$$

Step-by-Step Solution

Answer

The product is -(18a^3 b^3).

Step 1: Multiply the Signs

First, consider the signs of each term in the product. We have three negative signs: - The product of two negative numbers is positive. For each pair of negative terms, the result will be positive. - The product of all three negative numbers will be negative since a negative times a positive is negative. Thus, the result of multiplying i.e. (-a b)(-3 a b)(-6 a b) = - (a^3 b^3). Overall, the product of the signs is negative.

Step 2: Combine Coefficients

Next, multiply the numerical coefficients: 1 (from a b) imes 3 imes 6 = 18. So, the coefficient for the product will be 18.

Step 3: Multiply Variables

Since a b is squared through the multiplication (a b)(a b)(a b), combine like terms: a^1 imes a^1 imes a^1 = a^3 and, b^1 imes b^1 imes b^1 = b^3. Thus, combining these terms gives us: a^3 b^3.

Step 4: Combine All Parts

Combine the coefficient and the powers computed from the previous steps. The final product is: - (18) a^3 b^3.

Key Concepts

Understanding Negative Numbers in Polynomial Multiplication

Negative numbers can sometimes be a bit tricky in multiplication. A key rule to remember is that multiplying two negative numbers gives a positive result. However, if you multiply three negative numbers, the product becomes negative again. This happens because the first two negatives cancel each other out (turning positive), but the third negative flips the product back to negative.

In our exercise, we dealt with three negative terms:

• each negative sign ultimately makes the entire product negative because a negative times a positive remains negative.

Understanding these basic interactions of negative numbers is crucial when working with polynomial expressions.

Importance of Coefficients in Polynomial Multiplication

Coefficients are the numbers that multiply the variables in an algebraic term. When multiplying polynomials, combining coefficients is a straightforward step, but it's very important to do it correctly.

For example, in the problem

• the coefficients are 1 (from each a b), -3, and -6.

To find the coefficient of the resulting product, we multiply these numbers. Remember to apply the rules of negative number multiplication here; multiplying

• 1 × 3 × 6 gives 18,

and the sign is already determined to be negative from the previous section's explanation of negative numbers.

Thus, the resulting coefficient is -18. Whenever dealing with polynomials, paying attention to coefficients ensures you get the right numerical part of the answer.

Variables and Their Role in Multiplication

Variables in polynomials are symbols like \(a\) and \(b\) that represent numbers, and in multiplication, these symbols are treated as factors. The rules of exponents apply when multiplying variables, which are fundamental in simplifying polynomial expressions.

In the exercise, we multiplied

• a beach time, first with itself, then again, leading to the expression \((a b)(a b)(a b)\).

To combine these, recognize that each occurrence of a variable multiplies with another of the same kind.

• For \(a\), a^1 \times a^1 \times a^1 = a^3.
• For \(b\), b^1 \times b^1 \times b^1 = b^3.

This process illustrates how variables are handled similarly to numbers through understanding and application of exponents.

The Power of Exponents in Expressions

Exponents are shorthand notation for repeated multiplication. They play a significant role in simplifying and managing polynomial expressions. For example, when you encounter

• a^1 \times a^1 \times a^1,

you use the property of exponents where the powers are added:

• a^1 + a^1 + a^1 = a^3.

Similarly,

• b^1 \times b^1 \times b^1 = b^3.

Specifically, these concepts allow one to ascertain that the variables are multiplied correctly, resulting in a compact, easily readable final expression. Mastering exponents improves understanding and efficiency in managing polynomials, as observed in the multiplication step of this problem where we derived \(-18a^3b^3\). This way, the expression is neatly simplified into a combined format.