Problem 5 - Algebra for College Students

Question

Find each product. $$\left(-a^{2} b\right)\left(-4 a b^{3}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Multiply the Coefficients

Identify the numerical coefficients of each term. In the expression \((-a^2b)(-4ab^3)\), the coefficients are \(-1\) and \(-4\). Multiplying \(-1\) and \(-4\) gives \(4\).

Step 2: Multiply the a Terms

Now focus on the powers of \(a\). The expression has \(a^2\) in the first term and \(a\) in the second term. To multiply these, add the exponents: \(a^{2+1} = a^3\).

Step 3: Multiply the b Terms

Next, consider the powers of \(b\). The expression has \(b\) in the first term and \(b^3\) in the second term. When multiplying, add the exponents: \(b^{1+3} = b^4\).

Step 4: Combine the Results

Combine the results from Steps 1, 2, and 3. The product of the entire expression is the coefficient from Step 1 followed by the variables with their respective powers from Steps 2 and 3. This gives us \(4a^3b^4\).

Key Concepts

Understanding Coefficient Multiplication

When working with algebraic expressions, coefficient multiplication is one of the initial steps. The coefficients are the numerical parts of terms in an expression. For example, in the expression \(-a^2b\), the coefficient is \(-1\), and in \(-4ab^3\), it is \(-4\). When multiplying these coefficients, simply multiply the numbers: \(-1 imes -4\). Remember that a negative multiplied by a negative results in a positive. So, \(-1 imes -4 = 4\). This straightforward step helps simplify the multiplication of algebraic terms efficiently.

Applying Exponent Rules

Exponent rules come into play when multiplying like bases, an essential process in algebra. For example, consider multiplying \(a^2\) with \(a\). The base here is \(a\), and we use the rule: when multiplying like bases, add their exponents. Therefore, \(a^2 imes a = a^{2+1} = a^3\). Similarly, for the terms involving \(b\), you multiply \(b\) by \(b^3\), giving you \(b^{1+3} = b^4\).

These rules significantly streamline the process of simplifying expressions containing powers, avoiding the lengthy expansion needed otherwise.

Combining Polynomial Multiplication

Polynomial multiplication involves multiplying each term in one polynomial by every term in the other. In the problem \[(-a^2 b)(-4 a b^3)\], you begin by separately multiplying coefficients, and then apply exponent rules for the variables.

Here's a quick recap:

• Multiply \(-1\) and \(-4\) to get \(4\).
• Multiply \(a^2 imes a\) to get \(a^3\).
• Multiply \(b imes b^3\) to get \(b^4\).

After these steps, you assemble the entire expression, giving a final product of \(4a^3b^4\).

Mastering polynomial multiplication simplifies otherwise complex expressions and is a cornerstone skill in algebra.