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Chapter 10: Problem 56

Subtract \(\left(-5 w^{2}+6 w-3\right)\) from \(\left(11 w^{2}-20 w+4\right)\).

Short Answer

Expert verified
The result is \(16 w^{2} - 26 w + 7\).

Step by step solution

01

Write Down the Original Expressions

Identify and write down the original polynomials. The two polynomials are 1) \((11 w^{2} - 20 w + 4)\) and 2) \((-5 w^{2} + 6 w - 3)\).
02

Set Up the Subtraction

Since the task is to subtract the second polynomial from the first, set up the subtraction as follows: \((11 w^{2} - 20 w + 4) - (-5 w^{2} + 6 w - 3)\).
03

Distribute the Negative Sign

Distribute the negative sign to each term in the second polynomial: \((11 w^{2} - 20 w + 4) + 5 w^{2} - 6 w + 3\).
04

Combine Like Terms

Combine like terms to simplify the expression:- Combine the \(w^2\) terms: \(11 w^{2} + 5 w^{2} = 16 w^{2}\).- Combine the \(w\) terms: \(-20 w - 6 w = -26 w\).- Combine the constant terms: \(4 + 3 = 7\).The simplified result is: \(16 w^{2} - 26 w + 7\).

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Key Concepts

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

Combining Like Terms
When working with polynomials, it's important to combine like terms to simplify the expression. Like terms are terms that have the same variable and the same exponent. For instance:
- In the problem, we had \(11 w^2\) and \(-5 w^2\), both of which are 'like terms' because they contain \(w^2\).
- The coefficients (numbers in front) of these terms are added together.
So, \(11 w^2 + 5 w^2 = 16 w^2\).
Similarly, the terms with plain \(w\) (like \(-20 w\) and \(-6 w\)) should be combined:
\(-20 w - 6 w = -26 w\).
Combining like terms helps to simplify polynomials and make them easier to work with.
Distributive Property
The distributive property is an essential algebraic tool used when working with polynomials. It states that \(a(b + c) = ab + ac\).
In polynomial subtraction, the distributive property helps to remove the parentheses:
- In our case, we needed to subtract the polynomial \((-5 w^2 + 6 w - 3)\) from \((11 w^2 - 20 w + 4)\).
- On setting up the subtraction, we write it as: \( (11 w^2 - 20 w + 4) - (-5 w^2 + 6 w - 3)\).
- By distributing the negative sign across all terms inside the parentheses, we get:
\(11 w^2 - 20 w + 4 + 5 w^2 - 6 w + 3\).
This step is crucial before combining any like terms, ensuring the expression is correctly simplified.
Simplifying Expressions
Simplifying expressions is the process of making the expression as concise as possible. This includes removing parentheses, applying the distributive property, and combining like terms.
For our problem, after distributing the negative sign and combining like terms, we get:
- The \(w^2\) terms: \(11 w^2 + 5 w^2 = 16 w^2\).
- The \(-20 w\) and \(-6 w\) combine to make: \(-26 w\).
- And finally, combining the constant terms, we have \(4 + 3 = 7\).
Combining all, we achieve the simplified polynomial: \(16 w^2 - 26 w + 7\).
Simplified expressions are easier to read, interpret, and solve any further mathematical operations or equations.

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