Problem 108 - Algebra and Trigonometry

Question

In Exercises 107–110, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ (x-5)^{2}=x^{2}-5 x+25 $$

Step-by-Step Solution

Answer

The statement \( (x-5)^{2} = x^{2}-5x+25 \) is False. The correct statement is \( (x-5)^{2} = x^{2}-10x+25 \)

Step 1: Identify the Formula

For any binomial \( (a-b)\), when it is squared it expands to \( a^2 - 2ab +b^2 \)

Step 2: Apply the Formula

In the given equation substitute \( a =x \) and \( b =5\) into the formula. Doing this will give \( x^2- 2*5x+25\), simplified to \( x^2- 10x + 25\).

Step 3: Analyze the Statement

Comparing the statement \( x^2- 10x + 25\) from Step 2 with the provided statement \( x^2 -5x + 25\), it is evident that they are not equal. Therefore, the original statement is incorrect.

Step 4: Correct the Statement

To correct the original statement, replace \( -5x\) in the original statement with \( -10x\). Doing this will make the statement read \( (x-5)^2 = x^2 -10x +25 \) which is correct according to the binomial square formula.

Key Concepts

Polynomial Identity

A polynomial identity is an equation that holds true for all values of the variables involved. These identities are crucial in algebra as they allow us to simplify and manipulate complex expressions efficiently. One common type of polynomial identity is the square of a binomial, which has the general formula

• \((a-b)^2 = a^2 - 2ab + b^2\)

This identity is essential when working with quadratic expressions involving binomials. To understand this identity, it's helpful to recognize that it represents not just an equation, but a comprehensive rule governing how binomials behave when squared.
By knowing and applying polynomial identities, you can quickly verify the correctness of algebraic expressions and make concise corrections to prove or adjust statements. This concept forms a foundation in algebra that underlies many proofs and problem-solving techniques.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations such as addition, subtraction, multiplication, and division. In expressions, variables represent unknown or changeable values. A simple example of an algebraic expression is:

• \(3x + 2\)

Here, \(3x\) suggests 3 times an unknown value \(x\), and the \(+ 2\) indicates an additional two units.When simplifying or manipulating algebraic expressions, recognizing patterns, like those in polynomial identities, becomes highly valuable. For instance, when evaluating the statement

• \((x - 5)^2 = x^2 - 10x + 25\)

you need to apply your understanding of the binomial square identity to ensure each part of the expression is calculated correctly. If you overlook a factor like the \(2ab\) term in the identity, it can lead to incorrect simplifications, as seen in the incorrect step of using \(-5x\) instead of \(-10x\).
Practicing the manipulation of algebraic expressions strengthens algebra skills, enabling more efficient problem solving and easier verification of your results.

Mathematical Proof

Mathematical proof is a logical argument that establishes the truth of a given statement unequivocally. In algebra, proofs often involve verifying that an equation or identity holds under all conditions for the variables in play. The act of proving contributes to a deeper understanding of the concepts at hand, whether simple or complex.In the discussed exercise, we are proving that

• \((x - 5)^2 = x^2 - 10x + 25\)

by expanding the binomial as per known identities and comparing terms. Each step of the proof should logically follow from the previous one, systematically showing how both sides of the equation match under the rules of arithmetic and algebraic manipulation.
A mathematical proof requires that every element of the statement be justified; this meticulous process not only cements the truth of the identity but also reinforces the solver's mastery of algebraic principles. Through practicing proofs, one builds a robust conceptual framework that aids in tackling increasingly complex mathematical challenges.