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Chapter 12: Problem 64

Review factoring polynomials (Sections \(6.1-6.6)\) Factor $$ x^{2}-20 x+100 $$

Short Answer

Expert verified
\( x^2 - 20x + 100 = (x - 10)^2 \)

Step by step solution

01

- Identify the Polynomial

The given polynomial is a quadratic expression of the form: \[ x^2 - 20x + 100 \].
02

- Recognize the Perfect Square Trinomial

Notice that the terms resemble the structure of a perfect square trinomial: \[ a^2 - 2ab + b^2 \].
03

- Determine the Values of a and b

Here, identify \(a = x\) and \(b = 10\). This is because: \[ x^2 - 2(x)(10) + 10^2 \].
04

- Write as a Square of a Binomial

Since the polynomial fits the pattern \( x^2 - 2(x)(10) + 10^2 \), it can be rewritten as: \[ (x - 10)^2 \].

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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 2, which means the highest exponent of the variable is 2. The standard form of a quadratic expression is \[ ax^2 + bx + c \]. Here:
  • a, b, and c are known as coefficients
  • x is the variable
Let's break this down:
  • \[ ax^2 \]: The term with the variable squared
  • \[ bx \]: The term with the variable raised to the first power
  • \[ c \]: The constant term with no variable involved

In our given polynomial, \[ x^2 - 20x + 100 \], we can see that
  • \[ a = 1 \]
  • \[ b = -20 \]
  • \[ c = 100 \]

Identifying these parts helps us when we need to factor the quadratic expression or solve it using various methods.
Perfect Square Trinomial
A perfect square trinomial is a special form of a quadratic expression. It turns into a square of a binomial. Perfect square trinomials have the form:
  • \[ a^2 + 2ab + b^2 \]

    • or
  • \[ a^2 - 2ab + b^2 \]

    • This means we recognize it by:
  • The first term being a square: \[ a^2 \]
  • The last term being another square: \[ b^2 \]
  • The middle term being twice the product of the two square roots: \[ 2ab \] or \[ -2ab \]
In the given exercise, \[ x^2 - 20x + 100 \], we can see:
  • \[ a = x \]
  • \[ b = 10 \]
Thus, it matches the form \[ (x)^2 - 2(x)(10) + (10)^2 \], which confirms it as a perfect square trinomial.
Binomial Square
A binomial is simply an algebraic expression containing two terms. When you square a binomial, we get a perfect square trinomial. This process goes as follows:
  • \[ (a + b)^2 = a^2 + 2ab + b^2 \]
  • \[ (a - b)^2 = a^2 - 2ab + b^2 \]
In our exercise, after recognizing the perfect square trinomial, it can be rewritten as: (x - 10)^2.
Here鈥檚 the breakdown:
  • \[ a = x \]
  • \[ b = 10 \]
Now, reversing it by expanding using the binomial formula, we get:
  • \[ (x - 10)^2 = x^2 - 2(x)(10) + 10^2 = x^2 - 20x + 100 \]
Therefore, the polynomial \[ x^2 - 20x + 100 \] neatly factors into the square of a binomial: \[ (x - 10)^2 \]. Understanding how to square a binomial or recognize one can make factoring polynomials much easier.

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Most popular questions from this chapter

Express as an equivalent expression that is a sum or a difference of logarithms and, if possible, simplify. $$\log _{a} \frac{c-d}{\sqrt{c^{2}-d^{2}}}$$

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Use the properties of logarithms to find each of the following. $$\log _{3} 27^{7}$$

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