Problem 4 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$-7 a b^{2}\left(2 b^{3}-3 a^{2}\right)$$

Step-by-Step Solution

Answer

The product is \\(-14ab^5 + 21a^3b^2\\).

Step 1: Apply the distributive property

To find the product, we need to distribute the expression outside the parenthesis, \(-7ab^2\), across each term within the parenthesis: \(2b^3 - 3a^2\). This means we'll multiply \(-7ab^2\) with \(2b^3\) first, and then with \(-3a^2\).

Step 2: Multiply first term

First, multiply \(-7ab^2\) by \(2b^3\): \(-7ab^2 \times 2b^3 = -14ab^{2+3} = -14ab^5\).

Step 3: Multiply second term

Next, multiply \(-7ab^2\) by \(-3a^2\): \(-7ab^2 \times -3a^2 = 21a^{1+2}b^2 = 21a^3b^2\). Notice how the negative signs cancel each other out, turning the expression positive.

Step 4: Combine the terms

The final result is the sum of the products from steps 2 and 3: \(-14ab^5 + 21a^3b^2\). This is the expanded form of the original expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies expressions by removing parentheses and distributing a factor across terms inside parentheses. This property is especially useful in polynomial multiplication where expressions consist of more than one term. To apply the distributive property, each term inside the parenthesis is multiplied by the term outside.

For the given problem, \[-7ab^2(2b^3 - 3a^2)\], the practicable steps included the multiplication of each term inside the parenthesis by \(-7ab^2\).

• Multiply \(-7ab^2\) by \(2b^3\).
• Multiply \(-7ab^2\) by \(-3a^2\).

Each of these steps ensures that every term in the parenthesis is accounted for with the multiplier \(-7ab^2\), resulting in two separate products that are then summed together. This approach helps not only in simplifying expressions but also in solving equations.

Polynomial Multiplication

Polynomial multiplication involves the process of multiplying two polynomials, which can contain one or more terms. The goal is to expand the polynomial expressions by applying multiplication rules, combining like terms, and uncovering the resulting polynomial.

In our exercise, the original expression \(-7ab^2(2b^3 - 3a^2)\) represents a single monomial multiplied by a binomial. To achieve polynomial multiplication, each term of the binomial \(2b^3\) and \(-3a^2\) is multiplied by the monomial \(-7ab^2\).

• First product: \(-7ab^2 \times 2b^3 = -14ab^5\).
• Second product: \(-7ab^2 \times -3a^2 = 21a^3b^2\).

The results are combined to generate the expanded form of \(-14ab^5 + 21a^3b^2\). This multiplication process effectively transforms the original expression into a polynomial of greater degree.

Binomial Expressions

Understanding binomial expressions is crucial when tackling problems that involve polynomial multiplication. A binomial is a type of polynomial with exactly two terms. In this exercise, the binomial is \(2b^3 - 3a^2\). Each term represents a different part of the expression that needs to be multiplied by the same monomial \(-7ab^2\).

Remember, when multiplying binomials by other terms or expressions, it’s essential to consider the structure:

• Identify the terms in the binomial (in our case: \(2b^3\) and \(-3a^2\))
• Apply the distributive property, one term at a time

This approach ensures the calculation of each term is meticulously addressed, minimizing the risk of errors and maximizing understanding of the multiplication's impact.