Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ -x^{3}+2 x\left(x-x^{2}\right) $$

Short answer

The simplified version of the given equation is \(-2x^4-x^3+2x^2\).

Step-by-step solution

Distribute the Multiplier to the Objects within the Parentheses

Distribute the multiplier, which is 2x, into the binomial bracket, i.e., multiply each term inside the bracket with 2x:\n\n \(-x^3+2x(x)-2x(x^2)\)

Simplification

First, reduce the equation by carrying out the multiplication: \n\n \(-x^3+2x^2-2x^4\)

Rearrange and Combine Like Terms

Sort the terms based on the powers of \(x\) (from the highest to the lowest) and combine the like terms. In this case, there are no like terms, but rearranging gives us a more standard form of the polynomial. \n\n \(-2x^4-x^3+2x^2\).

Key concepts

The Distributive Property

Understanding the distributive property is essential in algebra, especially when simplifying expressions involving parentheses. This property is expressed mathematically as: \( a(b + c) = ab + ac \). Essentially, it states that if you have a term outside a set of parentheses (the 'distributor'), you distribute it by multiplying it with each term inside the parentheses.

Let's apply this in our exercise example. We start with the expression \( -x^{3}+2x(x-x^{2}) \). Applying the distributive property, we multiply \( 2x \) by each term inside the parentheses: \( 2x \times x \) and \( 2x \times -x^{2} \). This yields \( -x^{3}+2x^{2}-2x^{4} \), where we have successfully distributed \( 2x \) across the binomial \( (x-x^{2}) \).

When using the distributive property, ensure that each term within the parentheses is multiplied by the term outside, taking care to correctly apply signs (positive or negative) as they are critical to obtaining the correct simplification of the expression.

Simplifying Expressions

Simplifying expressions is a core skill in algebra. This process often involves combining like terms and applying various algebraic properties, such as the distributive property. Like terms are terms that have exactly the same variable parts, which means they have the same variable raised to the same power. Coefficients of these terms can be different, but to combine them, you simply add or subtract the coefficients as appropriate.

For instance, in our exercise, after the distributive step, we obtain \( -x^{3}+2x^{2}-2x^{4} \). To simplify this, you need to identify like terms. In this case, there are no like terms to combine. However, simplifying often includes arranging terms in descending order of their exponents, a convention called the standard form of a polynomial. Hence, the simplified expression becomes \( -2x^4-x^3+2x^2 \).

Polynomials

Polynomials are algebraic expressions consisting of variables and coefficients, which are combined using addition, subtraction, multiplication, and non-negative integer exponents. Examples include \( 2x + 3 \) or \( -x^3 + 4x^2 - 7 \). The degree of a polynomial is determined by the highest power of the variable in the expression.

In our problem, \( -2x^4-x^3+2x^2 \) is a polynomial of degree 4 because the highest exponent of \( x \) is 4. It is essential when handling polynomials to ensure that terms are arranged in descending order to reflect the standard form, making it easier to identify the leading term (the term with the highest power of the variable) and to perform operations such as addition, subtraction, or division of polynomials.