Question

In Exercises 25 and 26, describe and correct the error in performing the operation. \(\begin{aligned}(2 x-7)^3 &=(2 x)^3-7^3 \\ &=8 x^3-343 \end{aligned}\)

Short answer

The correct way to expand \( (2x-7)^3 \) according to the power rule is \( (8x^3 + 84x^2 - 294x - 343) \).

Step-by-step solution

Correct Application of Binomial Power Rule

According to the power rule, the cube of a binomial \( (a+b)^3 \) equals \( a^3 + 3a^2b + 3ab^2 + b^3 \). Apply this rule to the binomial \( (2x-7)^3 \) so that it becomes \( (2x)^3 - 3(2x)^2*(-7) + 3*(2x)*(-7)^2 + (-7)^3 \).

Simplify the Result

After the first step, we have \( (8x^3 + 84x^2 - 294x - 343) \). Simplify this further to its simplest form.

Key concepts

Polynomial Operations

Polynomial operations involve various arithmetic processes applied to polynomials. Polynomials are expressions consisting of variables and coefficients, connected by operations such as addition, subtraction, and multiplication. Simple operations can include adding or subtracting polynomials by combining like terms. However, multiplying polynomials requires distributing each term in one polynomial to every term in another.

For multiplication, such as expanding a binomial raised to a power, we use more specific operations like applying the Binomial Theorem. Orders of operations are crucial here. Remember to calculate powers of individual terms before applying multiplication across different terms.

In the context of the given exercise, performing polynomial operations incorrectly can lead to substantial errors. It's essential to follow each operation step closely, keeping track of all terms and signs.

Binomial Expansion

The Binomial Theorem is a powerful tool for expanding expressions of the form \( (a + b)^n \). It states that such an expression can be expanded as the sum of terms of the form \( C(n, k) a^{n-k} b^k \), where \( C(n, k) \) is a binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \).

For small powers, the binomial expansion can be manually calculated by breaking down the expression and applying the necessary coefficients. In the exercise, \( (2x - 7)^3 \) was expanded.

• The first term is \( (2x)^3 \). Calculating gives \( 8x^3 \).
• The second term needs three times the product of \( (2x)^2 \) and \( -7 \), resulting in \( 84x^2 \).
• The third term uses three times the product of \( 2x \) and \( (-7)^2 \), giving \( 294x \).
• The last term is \( (-7)^3 \), yielding \( -343 \).

Expanding step by step and adhering to the theorem ensures the expression is correctly solved.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding how to manipulate them is fundamental in algebra. They can represent real-world quantities and can be simplified to solve equations.

Expressions such as \( 2x - 7 \) are building blocks in algebra. When raised to a power, as in the exercise, processes such as the binomial theorem are applied to simplify or expand.

• Variables like \( x \) represent unknown values that can be manipulated through algebraic rules.
• Operations include basic arithmetic but can also involve more complex manipulations like factoring or expansion.
• Arming yourself with formulas and rules, such as binomial expansion, helps evaluate expressions effectively.

Emphasizing precision in calculations and a clear understanding of variable manipulation ensures successful problem-solving in algebra.