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Chapter 6: Problem 72

In Exercises 67-74, factor the polynomial completely. $$ x^{2}-22 x+121 $$

Short Answer

Expert verified
The factorized form of the given quadratic equation \(x^{2}-22x+121\) is \((x-11)^2\).

Step by step solution

01

Identifying 'Perfect Square Trinomials'

A 'Perfect Square Trinomial' is a trinomial that is the square of a binomial. In other words, it can be written in the form \((a+b)^{2}\) or \((a-b)^{2}\). Considering the equation \(x^{2}-22x+121\), it appears to be the square of a binomial.
02

Factoring the Polynomial

Let's rewrite the expression \(x^{2}-22x+121\) in the form \((a-b)^{2}\). This factorization results in \((x-11)^{2}\) where a=x and b=11.

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Key Concepts

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

Understanding Perfect Square Trinomials
Perfect square trinomials are special polynomial expressions that can be rewritten as the square of a binomial. They typically have the general form of \(a^2 \pm 2ab + b^2\). This means when you take a binomial, such as \(a \pm b\), and square it, you will end up with a perfect square trinomial.
To identify a perfect square trinomial, you need to check that:
  • The first term is a perfect square, \(a^2\).
  • The third term is a perfect square, \(b^2\).
  • The middle term is twice the product of the square roots of the first and the third terms, \(2ab\).
For example, the polynomial \(x^2 - 22x + 121\) is a perfect square trinomial because:
  • The first term, \(x^2\), is a perfect square.
  • The third term, \(121\), is a perfect square \((11^2)\).
  • The middle term, \(-22x\), is twice the product of \(x\) and \(11\) (since \(-2 \cdot 11 \cdot x = -22x\)).
Recognizing these patterns helps in simplifying the factoring process.
The Simplicity of Binomial Squares
A binomial square is formed when a binomial expression, like \( (a + b) \) or \( (a - b) \), is squared. The result is a perfect square trinomial.Consider the binomial \((a + b)^2\), which will expand to \(a^2 + 2ab + b^2\). Similarly, \((a - b)^2\) will expand to \(a^2 - 2ab + b^2\).
Factoring involves working backwards from this expansion. If you start with a trinomial like \(x^2 - 22x + 121\), notice it resembles the structure \((x - 11)^2\) because:
  • The first term, \(x^2\), matches \(a^2\).
  • The last term, \(121\), matches \(b^2\) with \(b = 11\).
  • The middle term, \(-22x\), fits \(-2ab\), where \(a = x\) and \(b = 11\).
Thus, you factor the trinomial to \((x - 11)^2\), revealing it as a binomial square.
Polynomial Factorization in Practice
Polynomial factorization is the process of breaking down a complex polynomial into simpler polynomials that, when multiplied together, result in the original polynomial. It serves to simplify expressions and solve polynomial equations more easily.
Here are steps for effective polynomial factorization:
  • Identify if the polynomial is a recognizable form, like a perfect square trinomial.
  • Look for common factors across terms.
  • Use special formulas and identities, such as the difference of squares or perfect square trinomials.
For the polynomial \(x^2 - 22x + 121\), you can quickly recognize it as a perfect square trinomial. By knowing the pattern \((a - b)^2 = a^2 - 2ab + b^2\), you immediately factor it into \((x - 11)^2\).
Recognizing and using these patterns makes polynomial factorization less daunting and much quicker.

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Most popular questions from this chapter

The figure below shows two cubes: a large cube whose volume is \(a^{3}\) and a smaller cube whose volume is \(b^{3}\). If the smaller cube is removed from the larger, the remaining solid has a volume of \(a^{3}-b^{3}\) and is composed of three rectangular boxes, labeled Box 1 , Box 2, and Box 3. Find the volume of each box and describe how these results are related to the following special product pattern. $$ \begin{aligned} a^{3}-b^{3} &=(a-b)\left(a^{2}+a b+b^{2}\right) \\ &=(a-b) a^{2}+(a-b) a b+(a-b) b^{2} \end{aligned} $$

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