Problem 153 - College Algebra

Question

Perform the indicated operation or operations. $$ (2 x-1)\left(x^{2}+x-2\right) $$

Step-by-Step Solution

Answer

The result of the multiplication is \(2x^3 + x^2 - 5x + 2\)

Step 1: Distribute the First Term of the Binomial

The first term of the binomial (2x) is multiplied by each term in the quadratic binomial. So, \(2x * x^2\) gives \(2x^3\), and \(2x * x\) gives \(2x^2\), and \(2x * -2\) gives \(-4x\). The partial result is \(2x^3 + 2x^2 - 4x\)

Step 2: Distribute the Second Term of the Binomial

The second term of the binomial (-1) is now multiplied by each term in the quadratic binomial. So, \(-1 * x^2\) gives \(-x^2\), and \(-1 * x\) gives \(-x\), and \(-1 * -2\) gives \(2\). The partial result is \(-x^2 - x + 2\)

Step 3: Combine the Results from Steps 1 and 2

Now, add the results from Steps 1 and 2 to get final polynomial. This gives: \(2x^3 + 2x^2 - 4x - x^2 - x + 2\)

Step 4: Simplify Final Polynomial

Simplify the polynomial by combining like terms, that is terms with same degree. This leads to the final answer: \(2x^3 + x^2 - 5x + 2\)

Key Concepts

Distributive Property

Understanding the distributive property is essential when multiplying polynomials. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. When dealing with polynomials, like in the exercise \( (2x-1)(x^2+x-2) \), it’s the distributive property that allows us to multiply each term in one polynomial by every term in the other.

Think of it as distributing the individual terms of the first polynomial (known as the binomial in this case) across the terms of the second polynomial. The process involves two key steps: multiply each term of the binomial by every term of the quadratic polynomial, and then add the resulting polynomials together. The distributive property ensures that all important interactions between the polynomials are accounted for.

Combining Like Terms

After utilizing the distributive property, we end up with an expanded polynomial consisting of several terms. To simplify it, we need to combine like terms. Like terms are terms that have the exact same variable raised to the same power, albeit potentially with different coefficients.

Why is this important? It's because only like terms can be combined through addition or subtraction, which simplifies the polynomial and makes it easier to understand, solve, or use in further calculations. In our exercise result, \(2x^3 + 2x^2 - 4x - x^2 - x + 2\), the like terms that can be combined are \(2x^2\) and \( -x^2\), as well as \( -4x\) and \( -x\). Combining like terms is crucial for obtaining a simpler, more manageable expression.

Simplifying Polynomials

The process of simplifying polynomials is the act of reducing a complex polynomial expression to its simplest form. It involves the application of several algebraic rules, including the distributive property and the combination of like terms, as we've discussed already. Simplification is not just a matter of making the expression shorter or more elegant; it is about making the polynomial more practical to work with for further algebraic manipulations or for solving equations.

In our case, the initial multiplication results in \(2x^3 + 2x^2 - 4x - x^2 - x + 2\), a rather bulky polynomial. By combining like terms, we arrive at the much cleaner expression \(2x^3 + x^2 - 5x + 2\). Simplifying polynomials makes it possible to evaluate the polynomial more easily when substituting values for its variables, and it helps in understanding the structure of the polynomial.

Binomial Products

When multiplying two binomials, or a binomial and another polynomial as in our exercise, it's helpful to recognize certain patterns. A binomial is a type of polynomial with just two terms. The product of a binomial and another polynomial often results in a pattern that is characterized by its repeated, systematic approach—each term in the binomial is multiplied by each term in the other polynomial.

To successfully multiply binomials, one must apply the distributive property meticulously, ensuring that each term is accounted for. Our exercise \( (2x-1)(x^2+x-2) \) showcases a binomial being multiplied by a quadratic polynomial, thus resulting in a trinomial product. The ability to multiply binomials smoothly and accurately is foundational for students tackling algebra, as it is a frequent task in many different areas of mathematics.