Problem 11 - Algebra 1: Concepts and Skills

Question

Tell whether the statement is true or false. If the statement is false, rewrite the right-hand side to make the statement true. $$ (5 x-1)^{2}=25 x^{2}-10 x+1 $$

Step-by-Step Solution

Answer

The statement \( (5x-1)^{2}=25x^{2}-10x+1 \) is true.

Step 1: Identify the Binomial

The binomial given is \( (5x-1) \), and the statement is proposing that the square of this is equal to the right-hand side.

Step 2: Apply the Binomial Square Formula

The formula for the square of a binomial \( (a-b)^2 \) is \( a^2 - 2ab + b^2 \). If we substitute \(5x\) for \(a\) and \(1\) for \(b\) into this formula, we get \( (5x)^2 - 2*5x*1 + 1^2 = 25x^2 - 10x + 1 \).

Step 3: Comparing the Result with Statement

Comparing our result of \( 25x^2 -10x + 1 \) with the given statement, we find that they match, thus the statement is true.

Key Concepts

Binomial Theorem

The Binomial Theorem is an essential concept in algebra, allowing us to expand expressions that are raised to a power. Specifically, it gives us a formula to expand expressions in the form of \((a+b)^n\), where \(n\) is a positive integer. This theorem is particularly useful because it saves us time and effort compared to multiplying a binomial by itself repeatedly.
To understand the theorem better, consider it as a structured way to find each term in the expansion of a binomial expression. Each term includes coefficients known as binomial coefficients, calculated using combinations. For instance, the binomial coefficients can be found using \(\binom{n}{k}\), where \(n\) is the power the binomial is raised to, and \(k\) is the term number starting from 0.
The expansion forms a series of terms where the power of \(a\) decreases, and the power of \(b\) increases in each successive term. This pattern continues up to the \(n\)th power. This process of calculation is applied directly when expanding simple expressions like \((5x-1)^2\), as used in the exercise, by considering the formula for a squared binomial.

Polynomial Expressions

Polynomial expressions are sums of terms consisting of variables raised to whole number powers and multiplied by coefficients. They are fundamental in various areas of mathematics, including factoring, solving equations, and calculus. In our exercise, the expression \((5x-1)^2\) is polynomial due to its form, involving a variable raised to a power, 2 in this case.
A polynomial expression may have constant, linear, quadratic, cubic, or higher degree terms, depending on the highest power of the variable. Linear polynomials have the highest power of one, quadratics have the highest power of two, and so forth. When expanded, \((5x-1)^2\) translates to a quadratic polynomial \(25x^2 - 10x + 1\), where \(25x^2\) is the quadratic term, \(-10x\) is the linear term, and \(1\) is the constant term.
Understanding how to manipulate and expand these expressions is crucial for solving algebraic problems. Each term in a polynomial contributes to its degree, form, and the relationships it can represent. Therefore, mastering the expansion and simplifications of polynomial expressions builds a strong foundation in algebra.

Algebraic Identities

Algebraic identities are equations that are always true for any value of the variables involved. These identities provide shortcuts to simplify and evaluate algebraic expressions. They are especially handy when dealing with polynomial expansions or simplifications.
For instance, the identity for the square of a binomial is \((a-b)^2 = a^2 - 2ab + b^2\). This identity quickly lets us expand any binomial squared without manual multiplication. In the exercise, we use this identity to confirm the expansion of \((5x-1)^2\) into \(25x^2 - 10x + 1\).
Algebraic identities are powerful tools in simplifying calculations, making proofs, and solving equations by reducing complexity. Recognizing and applying these identities allow you to solve problems efficiently and can deepen your understanding of the inherent structure within algebraic expressions.