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Chapter 5: Problem 55

\(\left(8 b^{2}-\frac{7}{10} b+5\right)-\left(6 b^{2}-\frac{1}{10} b+3\right)\)

Short Answer

Expert verified
\ \ \ \[2b^2 - \frac{3}{5}b + 2\]

Step by step solution

01

- Distribute the negative sign

First, distribute the negative sign to each term inside the second set of parentheses: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[8b^2 - \frac{7}{10}b + 5 - 6b^2 + \frac{1}{10}b - 3\]
02

- Combine like terms

Group the like terms together to simplify: \ \ \ \ \ \ \ \[ (8b^2-6b^2) + \bigg( -\frac{7}{10}b+\frac{1}{10}b \bigg ) + (5-3)\] \ \ Combine the coefficients of like terms: \ \ \ \ 8b^2 - 6b^2 = 2b^2 \ \ \ -\frac{7}{10}b + \frac{1}{10}b = -\frac{6}{10}b = -\frac{3}{5}b \ \ 5-3=2.
03

- Write the simplified expression

Combine all the simplified terms: \ \ \ \ \ Complete expression will be \[2b^2 - \frac{3}{5}b + 2\]

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Key Concepts

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

Distributing Negative Sign
When you see a subtraction symbol in front of a polynomial, it's essential to distribute the negative sign to every term inside the parentheses.

For example, consider the expression \( (8b^2 - \frac{7}{10}b + 5) - (6b^2 - \frac{1}{10}b + 3) \).
When you distribute the negative sign from the second set of parentheses, each term changes its sign:
  • The \(+6b^2\) becomes \(-6b^2\).
  • The \( -\frac{1}{10}b \) becomes \(+\frac{1}{10}b \).
  • The \(+3\) becomes \(-3\).
After distributing the negative sign, the expression is now: \[ 8b^2 - \frac{7}{10}b + 5 - 6b^2 + \frac{1}{10}b - 3 \].
This step is crucial for accurate polynomial subtraction.
Combining Like Terms
The next step in simplifying our example is to combine like terms.
Like terms are terms that have the same variables raised to the same powers.
In the expression \[ 8b^2 - \frac{7}{10}b + 5 - 6b^2 + \frac{1}{10}b - 3 \], identify the groups of like terms:
  • \(8b^2\) and \(-6b^2 \)
  • \(-\frac{7}{10}b\) and \( \frac{1}{10}b \)
  • \(5\) and \(-3\)
Combine these terms one group at a time: \( 8b^2 - 6b^2 = 2b^2 \).
For the next group: \( -\frac{7}{10}b + \frac{1}{10}b = -\frac{6}{10}b = -\frac{3}{5}b \).
Lastly, for the constants: \(5 - 3 = 2\). Combining like terms is vital in reducing the expression's complexity.
Simplification of Expressions
The final step involves writing the simplified expression after combining all the like terms.
From the previous section, we combined the terms and found:
  • \(2b^2\)
  • \(-\frac{3}{5}b\)
  • and \(2 \)
To create the final, simplified expression, simply put these combined terms together: \[2b^2 - \frac{3}{5}b + 2 \].
Each step of the simplification process helps streamline the polynomial, making it more straightforward and easier to understand and use in further calculations.

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Most popular questions from this chapter

A rectangular parking lot is three times as long as it is wide. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

Problem: Simplify: \((5 x-2)(x-3)\) Incorrect Answer: \((5 x-2)(x-3)\) $$ \begin{aligned} &=5 x(x)+5 x(-3)-2 x-2(3) \\ &=5 x^{2}-15 x-2 x-6 \\ &=5 x^{2}-17 x-6 \end{aligned} $$

\(\left(y^{2}-81\right) \div(y-9)\)

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Explain why "difference of squares" is a good name for the pattern \((a-b)(a+b)=a^{2}-b^{2}\).
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