Problem 17 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (2 x-3)\left(x^{2}-3 x+5\right) $$

Step-by-Step Solution

Answer

The product of the given polynomials is \(2x^3 - 9x^2 - 15\)

Step 1: Distribute the terms

Multiply each term in the binomial \(2x - 3\) with every term in the binomial \(x^2 - 3x + 5\). This can be broken down as follows: - Multiply the first terms: \(2x * x^2 = 2x^3\)- Multiply the outer terms: \(2x * -3x = -6x^2\)- Multiply the inner terms: \(-3 * x^2 = -3x^2\)- Multiply the last term: \(-3 * 5 = -15\)

Step 2: Combine like terms

After distributing, we are left with four terms: \(2x^3\), \(-6x^2\), \(-3x^2\), and \(-15\). Combine the like terms, i.e., the terms having same power of variable \(x\). So, \(-6x^2\) and \(-3x^2\) can be combined.

Step 3: Express the final result

Add \(-6x^2\) and \(-3x^2\) to get \(-9x^2\). So, the final expression is \(2x^3 - 9x^2 - 15\)

Key Concepts

Binomial Products

When we talk about binomial products, we refer to the result of multiplying two binomials together. A binomial is a polynomial with just two terms, and when two binomials are multiplied, we use a method often referred to as the FOIL method, which stands for First, Outer, Inner, Last. This is a way of remembering to multiply each term in the first binomial by each term in the second binomial.

For instance, let's look at the example from the exercise where we multiply \( (2x - 3) \times (x^2 - 3x + 5) \). Here, the first binomial is \(2x - 3\), and the second is \(x^2 - 3x + 5\). Using the distributive property, we expand and multiply every term in the first binomial by every term in the second binomial, ensuring no terms are left out.

Visualizing the FOIL method

In visual terms, imagine placing one of the binomials on top and the other on the left side of a two-by-two grid. You then multiply the terms that intersect in each cell of the grid, which represents the multiplication of the First, Outer, Inner, and Last terms. This organized approach helps ensure that all possible products are considered.

Combining Like Terms

Once all the terms have been multiplied together using the distributive property, we are often left with what we call like terms. These are terms in an expression that have the exact same variable(s) raised to the same power. You can only combine, in other words, add or subtract like terms.

To combine like terms effectively, you begin by identifying terms that share the same variable and exponent. In our example \(2x^3 - 6x^2 - 3x^2 - 15\), the like terms are \( -6x^2 \) and \( -3x^2\). Once identified, we can add these terms together to simplify the polynomial.

Simplifying the Expression

By adding the like terms, we obtain \( -9x^2 \), hence simplifying the expression to \(2x^3 - 9x^2 - 15\). Combining like terms is critical for simplifying expressions and should be done with care to avoid any errors. It's a step that consolidates the multiplication's results into a more manageable and understandable form.

Distributive Property

The distributive property is a cornerstone of basic algebra and is used to multiply a single term across a sum or difference within parentheses. It rests on the principle that \( a(b + c) = ab + ac \). This property makes multiplying polynomials manageable because it allows us to distribute, or break down, the multiplication process into smaller, more straightforward steps.

In the exercise provided, the distributive property is applied when the term \(2x\) and \( -3\) from the first binomial are multiplied individually by each of the terms in the second binomial \(x^2\), \( -3x\), and \( 5\). Each of these individual multiplications gives us a term of the final product. When we use the distributive property, we ensure that each term in one polynomial is multiplied by each term in the other polynomial.

Orderly Multiplication

The distributive property allows us to handle polynomial multiplication in an orderly manner. By adhering to this property's rules, we can systematically approach complex expressions, making our work less prone to error and easier to follow.