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Chapter 5: Problem 23

Factor. $$t^{2}-2 t+1$$

Short Answer

Expert verified
The factorized form of \(t^{2} - 2t + 1\) is \((t - 1)^2\).

Step by step solution

01

Identify the coefficients and constant

The quadratic expression here is \(t^{2} - 2t + 1\). So, the coefficient of \(t^{2}\) is 1, the coefficient of t is -2, and the constant term is 1.
02

Find two numbers

To factorize this trinomial, we need to look for two numbers that add up to -2 (the coefficient of t) and multiply to 1 (the constant). After evaluating, we find that the two numbers that satisfy this are -1 and -1, because (-1) + (-1) = -2 and (-1) * (-1) = 1.
03

Rewrite the quadratic trinomial in factored form

We can then write the quadratic trinomial in factored form. The two pairs of brackets each have a 't' and are subtracted by the values we found in the previous step. So, \(t^{2} - 2t + 1\) can be rewritten as \((t - 1)(t - 1)\) or as \((t - 1)^2\)

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Key Concepts

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

Quadratic Expression
A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. It is commonly written in the standard form of ax^2 + bx + c, where a, b, and c are coefficients and a is not equal to zero.

The quadratic expression given in the exercise is t^2 - 2t + 1, where the variable t is squared, and the expression includes a linear term and a constant. This particular form is considered because it influences how we find the solutions to certain problems, especially in factoring.
Trinomial Factorization
Trinomial factorization involves breaking down a quadratic expression, specifically a trinomial, into a product of two binomials. The general approach includes finding two numbers that both add up to the middle term's coefficient and multiply to the constant term.

In our example, we analyzed the trinomial t^2 - 2t + 1 and searched for two numbers that add up to -2 and multiply to give 1. The numbers -1 and -1 fit these requirements. After this identification, we write the trinomial as (t - 1)(t - 1) or (t - 1)^2, which is its factored form. This step is crucial for simplifying equations or finding variable values that satisfy the original quadratic expression.
Coefficient Identification
In algebra, the process of coefficient identification involves determining the numerical factors that are multiplied by the variables in an expression. These coefficients play a pivotal role in various algebraic methods, including factoring.

In the provided example, we identify the coefficient of t^2 as 1, the coefficient of t as -2, and the constant term as 1. These values are not only part of the quadratic expression but they are also key in solving and simplifying the expression. By understanding the role of these coefficients, one can apply techniques like factoring more effectively to solve quadratic equations or to simplify algebraic expressions.

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