Problem 21 Factor. $$ x 2-2 x+1 $$... [FREE SOLUTION]

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Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 6: Problem 21

Factor. $$ x 2-2 x+1 $$

Short Answer

Expert verified

The expression factors to $(x-1)^2$.

Step by step solution

Identify the Quadratic

We need to factor the expression $x^2 - 2x + 1$. It is a quadratic in the form $ax^2 + bx + c$, where $a = 1$, $b = -2$, and $c = 1$.

Check if it's a Perfect Square

To check if the quadratic is a perfect square, we see if it can be expressed in the form $(x + d)^2$. Start by taking the square root of the first and last terms: $x$ and $1$. The middle term, $-2x$, should equal $2 * x * 1$ which it does. Therefore, $x^2 - 2x + 1$ is a perfect square.

Write as a Square

Since $x^2 - 2x + 1$ is a perfect square, it can be written as $(x - 1)^2$. This is because $x - 1$ multiplied by itself results in $x^2 - 2x + 1$.

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Key Concepts

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

Perfect Square

In mathematics, a perfect square is the product of an integer multiplied by itself. In terms of algebraic expressions, a perfect square quadratic can be expressed as

$(x + d)^2 = x^2 + 2dx + d^2$

This special form simplifies the factoring process because the expression results from squaring a binomial. When identifying a perfect square, you start by examining all terms of the quadratic to see if they fit the pattern of

$a^2$ as the first term,
$b^2$ as the third term,
and $2ab$ as the middle term.

For example, in the expression $x^2 - 2x + 1$,

$x^2 = (x)^2$
$1 = (1)^2$
and $-2x = 2 \cdot x \cdot (-1)$

This reveals that it can be expressed as

$(x - 1)^2$

This simplification makes problem solving quicker and easier. Recognizing these characteristics is crucial when dealing with quadratic equations.

Quadratic Expressions

A quadratic expression is a polynomial expression of degree two, typically written in the form

$ax^2 + bx + c$

where "a," "b," and "c" are constants, and "a" is not equal to zero. Quadratics are common in algebra because they describe a parabola in the coordinate plane.

The term with $x^2$ determines the parabola's direction (upwards if positive, downwards if negative).
The linear term $bx$ shifts the vertex horizontally.
The constant term "c" moves the parabola vertically.

In the specific case of the quadratic $x^2 - 2x + 1$, the leading term is $x^2$, indicating a parabola opening upwards. Understanding this structure helps in predicting graph shapes and factorization outcomes. These expressional shifts inform solutions beyond mere algebraic manipulation.

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. For quadratics, the focus is often on expressing them as a product of two binomials or identifying them as a perfect square. The importance of factoring lies in solving quadratic equations effectively.

It transforms a polynomial in its expanded form (e.g., $x^2 - 2x + 1$) into a factored form ($(x - 1)^2$).
When expressed in factored form, identifying roots or solutions is more straightforward, e.g., for $(x - 1)^2 = 0$, the solution is $x = 1$.

The approach often requires recognizing patterns (such as perfect squares) or using methods like grouping, difference of squares, or the quadratic formula. Gaining proficiency in factoring empowers students to tackle diverse mathematical problems more efficiently.

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91影视

Factor. $$ x 2-2 x+1 $$

Short Answer

Step by step solution

Identify the Quadratic

Check if it's a Perfect Square

Write as a Square

Key Concepts

Perfect Square

Quadratic Expressions

Polynomial Factoring

One App. One Place for Learning.

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