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Chapter 15: Problem 55

Add or subtract, as indicated. $$ \left(8 y^{2}-2 y-3\right)-\left(9 y^{2}-7 y-1\right) $$

Short Answer

Expert verified
\[-y^2 + 5y - 2\]

Step by step solution

01

- Distribute the Negative Sign

To subtract the second polynomial, distribute the negative sign to each term inside the parentheses. This gives: \[ (8y^2 - 2y - 3) - (9y^2 - 7y - 1) \] becomes \[ 8y^2 - 2y - 3 - 9y^2 + 7y + 1 \]
02

- Combine Like Terms

Now, combine the like terms from the resulting expression: Terms involving \(y^2\): \(8y^2 - 9y^2\) Terms involving \(y\): \(-2y + 7y\) Constant terms: \(-3 + 1\)
03

- Simplify Each Set of Like Terms

Simplify each set of like terms: Like terms involving \(y^2\): \[8y^2 - 9y^2 = -y^2\] Like terms involving \(y\): \[-2y + 7y = 5y\] Constant terms: \[-3 + 1 = -2\]
04

- Write the Final Simplified Expression

Combine all the simplified like terms together: \[-y^2 + 5y - 2\]

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Key Concepts

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

Distributing Negative Sign
When subtracting polynomials, it's crucial to distribute the negative sign across all terms of the polynomial that's being subtracted. This is because subtracting each term individually ensures accurate calculations. For example, consider the expression: \( (8y^{2} - 2y - 3) - (9y^{2} - 7y - 1) \). To distribute the negative sign, change the sign of every term inside the second set of parentheses.\
\
This changes the expression to: \[ 8y^2 - 2y - 3 - 9y^2 + 7y + 1 \].
\
Distributing the negative sign helps avoid mistakes by ensuring all negative values are applied correctly. This is an important first step in polynomial subtraction.
Combining Like Terms
Once the negative sign has been distributed, the next step is to combine like terms. Like terms have identical variables raised to the same power. In our example: \[ 8y^2 - 2y - 3 - 9y^2 + 7y + 1 \], we need to identify and combine the terms systematically:
\
- Terms involving \(y^2\): \(8y^2 - 9y^2\)
- Terms involving \(y\): \(-2y + 7y\)
- Constant terms: \(-3 + 1\)
Combining like terms consolidates multiple expressions into simpler forms, making it easier to solve or further manipulate. It's similar to gathering all similar items together before counting or weighing them.
Simplifying Expressions
After combining like terms, the final step is to simplify each set of like terms. This means performing the arithmetic operations on the grouped like terms. \
\
From the example: Terms involving \(y^2\): \(8y^2 - 9y^2 = -y^2\).\
\
- Terms involving \(y\): \(-2y + 7y = 5y\)
- Constant terms: \(-3 + 1 = -2\)
Once simplified, write the final expression by putting all simplified like terms together: \[-y^2 + 5y - 2 \]
Simplifying expressions reduces them to their most concise form, making it easier to understand and interpret. This clarity is important in both solving equations and communicating mathematical ideas clearly.

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