Problem 55 Find each product. $$(x-3)^{3}... [FREE SOLUTION]

Chapter 0: Problem 55

Find each product. $$(x-3)^{3}$$

Short Answer

Expert verified

The product of the expression $(x-3)^{3}$ is $x^{3} - 9x^{2} + 27x - 27$.

Step by step solution

Understand the exponent

Firstly, note that the exponent value is 3 which implies that the $(x-3)$ binomial expression is being multiplied by itself three times. This simplifies the expression to $(x-3) * (x-3) * (x-3)$

Multiplication of the first two

Now, multiply the first two $(x-3)$ binomial expressions. The multiplication for $x * x$ results in $x^2$. For $x * -3$ and $-3 * x$, both yield $-3x$ but when combined due to the like terms results in $-6x$. Lastly, the multiplication for $-3 * -3$ results in $9$. As a result, the new expression to simplify becomes $(x^2- 6x + 9) * (x-3)$

Final multiplication

Multiply the trinomial expression $x^2- 6x + 9$ with the binomial expression $(x-3)$. The multiplication for $x^2 * x$ results in $x^3$. For $x^2 * -3$, $-6x * x$ and $-3 * 9$, they result in $-3x^2$, $-6x^2$, and $-27$ respectively. Again, if like terms are combined, $ -3x^2 - 6x^2$ yields $-9x^2$. Thus, the final simplified expression becomes $x^{3} - 9x^{2} + 27x - 27$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Multiplication

Polynomial multiplication involves combining terms from two polynomial expressions to form a new polynomial. When dealing with binomials like $(x-3)$, we multiply the expressions by each other according to their exponents. For this problem, the binomial $(x-3)$ is taken to the power of 3, so it's multiplied by itself three times. This is done through a step-by-step approach:

Multiply the first two binomials: Start by expanding $(x-3) \times (x-3)$, producing a trinomial.
Multiply the result by the remaining binomial: Take the trinomial $(x^2 - 6x + 9)$ and multiply it by the third $(x-3)$ to obtain a polynomial.

By systematically combining terms and using the distributive property, a complex multiplication can be achieved seamlessly.

Exponentiation

Exponentiation in algebra refers to the process of raising a base (such as $(x-3)$) to a specific power. Here, the power of 3 implies that $(x-3)$ should be multiplied by itself, three times in total:

Repetition of multiplication: This means calculating $(x-3) \times (x-3) \times (x-3)$.
Understanding the size of the expression: Being raised to the third power is equivalent to cubing the expression, leading to a polynomial of degree three.

Exponentiation follows specific rules and allows for the systematic simplification and scaling of expressions, turning a simple binomial into a higher-degree polynomial.

Simplification of Algebraic Expressions

Simplifying algebraic expressions is about breaking them down into a less complicated form. It involves combining like terms and performing arithmetic operations to reach a cleaner result. In this exercise:

Combine like terms: As you multiply, like terms in polynomials will appear, such as $-6x^2$ and $-3x^2$. These combine to form $-9x^2$.
Simplify the expression: Each step consolidates the terms until reaching the final form, $x^3 - 9x^2 + 27x - 27$.

Effective simplification clarifies the expression, making it easier to understand and work with in further mathematical operations.

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91影视

Find each product. $$(x-3)^{3}$$

Short Answer

Step by step solution

Understand the exponent

Multiplication of the first two

Final multiplication

Key Concepts

Polynomial Multiplication

Exponentiation

Simplification of Algebraic Expressions

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