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Chapter 7: Problem 38

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(z-9)^{2}$$

Short Answer

Expert verified
(z - 9)^2 = z^2 - 18z + 81; it is a perfect square.

Step by step solution

01

Expand Using the Binomial Theorem

Recall the binomial theorem: \[ (a-b)^2 = a^2 - 2ab + b^2 \] Here, \( a = z \) and \( b = 9 \). Substitute these values into the formula.
02

Substitute the Given Values

Using \( a = z \) and \( b = 9 \), we get: \[ (z - 9)^2 = z^2 - 2 \times z \times 9 + 9^2 \]
03

Compute Each Term

Calculate each term step-by-step: \[ z^2 = z^2 \] \[ -2 \times z \times 9 = -18z \] \[ 9^2 = 81 \]
04

Combine the Results

Now combine all the computed terms together: \[ (z - 9)^2 = z^2 - 18z + 81 \]
05

Identify Special Pattern

The form \( (z - 9)^2 = z^2 - 18z + 81 \) is a perfect square trinomial.

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Key Concepts

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

Binomial Theorem
The binomial theorem is fundamental in algebra. It provides a way to expand expressions that are raised to a power, like \( (a-b)^2 \). The general form of the binomial theorem is: \[ (x + y)^n = \binom{n}{0} x^n + \binom{n}{1} x^{n-1} y + \binom{n}{2} x^{n-2} y^2 + \ldots + \binom{n}{n} y^n \].
This formula involves binomial coefficients, represented by \(\binom{n}{k}\). These coefficients can be found using combinations.
For our exercise, we used the simplified binomial expansion form: \[ (a - b)^2 = a^2 - 2ab + b^2 \],
which results from the binomial theorem applied to \( (a - b)^n \) when \( n = 2 \).
Trinomial Expansion
Trinomial expansion involves multiplying out expressions that result in a trinomial. In our case, \( (z - 9)^2 \) is expanded as:
\[ z^2 - 18z + 81 \]
Each term in this expansion comes from specific combinations of the terms in \( (z - 9) \).

This process uses the principles of the binomial theorem, specifically the identity: \[ (a - b)^2 = a^2 - 2ab + b^2 \].
It's important to methodically handle each pair of terms to ensure accuracy.
Algebraic Identities
Algebraic identities are equations that are true for any values of the variables. They help simplify and solve complex algebraic expressions. The identity used here is a perfect square trinomial:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]

Recognizing and using these identities makes solving algebra problems faster and easier. They also help understand the structure and properties of algebraic expressions. For example, knowing that \((z - 9)^2\) will always expand into: \[ z^2 - 18z + 81 \]
demonstrates the power and consistency of algebraic identities.
Difference of Squares
The difference of squares is another crucial algebraic identity represented as:
\[ a^2 - b^2 = (a + b)(a - b) \].
It's useful for factoring expressions and solving equations. However, it's important to note that our example: \((z - 9)^2\)
is not a difference of squares case. Instead, it is a perfect square trinomial.

Knowing the difference helps avoid mistakes and correctly apply the appropriate identity. Recognizing non-examples, like here, solidifies understanding.
Multiplication of Polynomials
Multiplying polynomials involves distributing each term in the first polynomial to every term in the second. With binomials, we use the FOIL method (First, Outer, Inner, Last). For \((z - 9)^2\),
we apply the FOIL method:
\[ (z-9)(z-9) = z(z-9) - 9(z-9) = z^2 - 9z - 9z + 81 \]
Thus:
\[ z^2 - 18z + 81 \].

This step-by-step ensures we combine terms correctly, resulting in the final expanded form.

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Most popular questions from this chapter

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you divide 2350 by \(48,600\), the result is closest to (a) 0.05 (b) 0.5 (c) 5 (d) 50 (e) 500

perform the indicated operation and simplify. $$\text { Solve for } x: \frac{x}{3}-\frac{x}{4}=3$$

Convert each number into scientific notation. $$56 \times 10^{-2}$$

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(1.93 \times 10^{5}\) by \(5.12 \times 10^{-3},\) the result is closest to (a) 10 (b) 1000 (c) \(10,000\) (d) 0.01 (e) 0.001
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