Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(3 x\left(x^{2}-2 x+1\right)\)

Short answer

The result of the operation is \(3x^{3} - 6x^{2} + 3x\)

Step-by-step solution

Identify the algebraic expressions to distribute

In the given expression \(3x(x^{2}-2x+1)\), \(3x\) needs to be distributed to every term inside the parentheses: \(x^{2}, -2x\), and \(1\).

Apply the distributive property

By the distributive property, multiply \(3x\) by each term inside the parentheses separately: \(3x * x^{2}\), \(3x * -2x\), and \(3x * 1\). Thus, the expression becomes \(3x^{3} - 6x^{2} + 3x\).

Write the resulting polynomial in standard form

The polynomial is in standard form when it is written with descending powers. Since the result of the previous step, \(3x^{3} - 6x^{2} + 3x\) is already in descending powers, it is in standard form.

Key concepts

Distributive Property

The distributive property is a crucial concept in mathematics, especially when dealing with algebraic expressions. This property is used to eliminate parentheses in expressions, which allows us to simplify and perform operations with ease. In algebra, the distributive property helps in multiplying a single term by several terms inside parentheses.

To apply it, you multiply the outside term by each term inside the parentheses separately. For example, given an expression like \(a(b + c)\), you use the distributive property to get \(ab + ac\). This ensures all parts of the expression are accounted for without changing its value.

• This property is often written as \(a(b + c) = ab + ac\).
• It allows for the simplification of expressions for further operations.
• It helps in expanding expressions during polynomial multiplication.

Understanding and applying this property makes solving polynomial expressions a lot simpler, as seen in the original exercise where \(3x\) is multiplied by each term in \((x^2 - 2x + 1)\).

Standard Form of Polynomial

The standard form of a polynomial is a way of writing polynomials such that the terms are ordered from the highest degree to the lowest degree. In practice, this means arranging the terms based on the exponents in a descending order.

When a polynomial is in standard form, it is easier to analyze and perform further operations like addition, subtraction, or even division. For any polynomial, this involves:

• Identifying the degree of each term.
• Reordering the terms so that those with the largest exponents are written first.
• Ensuring that the coefficients and variables are properly aligned in decreasing order of power.

By doing this, one can quickly understand and check the properties of a polynomial, such as its degree or leading coefficient. In the example solution, the final result \(3x^3 - 6x^2 + 3x\) is already presented in standard form.

Algebraic Expressions

Algebraic expressions are combinations of numbers and variables, entwined with operations like addition, subtraction, multiplication, and division. They form the basis of algebra and enable us to express mathematical relationships symbolically.

These expressions can range from simple to complex, incorporating constants, variables, exponents, and operators to form a meaningful representation of numbers. In essence, they are comprised of:

• Numerical coefficients, like the 3 in \(3x\).
• Variables, which are symbols that represent unknown quantities, such as \(x\).
• Operators, including +, -, and * which denote mathematical operations.
• Exponents, that indicate repeated multiplication of a variable, for example, \(x^2\).

Working with algebraic expressions involves combining and simplifying these elements using established rules, such as the distributive property, to solve equations or perform operations. The problem discussed in the original exercise perfectly illustrates how a complex expression can be simplified and organized into a standard polynomial form.