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Chapter 10: Problem 45

Multiply the polynomials. $$(7 z-1)^{2}$$

Short Answer

Expert verified
49z^2 - 14z + 1

Step by step solution

01

- Recognize the Structure

Notice that the polynomial is in the form \( (a - b)^2 \). Here, \( a = 7z \) and \( b = 1 \).
02

- Apply the Binomial Theorem

Use the binomial expansion formula \( (a - b)^2 = a^2 - 2ab + b^2 \).
03

- Substitute the Values

Substitute \( a = 7z \) and \( b = 1 \) into the formula. This gives \( (7z - 1)^2 = (7z)^2 - 2(7z)(1) + (1)^2 \).
04

- Simplify the Terms

Calculate each term separately: \( (7z)^2 = 49z^2 \), \( -2(7z)(1) = -14z \), and \( (1)^2 = 1 \).
05

- Combine the Terms

Combine the simplified terms to get the final expression: \( 49z^2 - 14z + 1 \).

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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 offers a formula for expanding expressions that are raised to a power, typically binomials. A binomial is an algebraic expression containing two terms, such as \( (a + b) \) or \( (a - b) \). The binomial theorem generalizes this to handle any power, resulting in a polynomial. For example, according to the binomial theorem:

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$\left(k^{-7}\right)^{2}$$

Perform the indicated operations. $$\left(2 x^{3}-5 x+8\right)-\left(4 x^{2}+2 x-3\right)+\left(-7 x^{3}+8 x^{2}\right)$$

Evaluate the polynomial for the given values of the variable. \(x^{2}+x+2\) a. for \(x=-10\) b. for \(x=\frac{2}{5}\)

Fill in the missing polynomial. $$(\quad)(w-7)=w^{2}-4 w-21$$

Simplify. Write the answers with positive exponents only. $$\frac{t^{4}}{t^{9}}$$
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