Problem 34 $$ \left(2 r^{3}+3 r s-5 s^{2}... [FREE SOLUTION]

Chapter 5: Problem 34

$$ \left(2 r^{3}+3 r s-5 s^{2}\right)-\left(5 r^{3}+r s+4 s^{2}\right) $$

Short Answer

Expert verified

-3r^{3} + 2rs - 9s^{2}

Step by step solution

- Distribute the negative sign

Expand the expression by distributing the negative sign to the second polynomial:\[ \text{First polynomial: } 2r^{3} + 3rs - 5s^{2} \]\[ \text{Second polynomial with distributed negative sign: } -5r^{3} - rs - 4s^{2} \]

- Combine like terms

Group and combine the like terms (terms with the same variables and exponents): \[ 2r^{3} - 5r^{3} + 3rs - rs - 5s^{2} - 4s^{2} \]

- Simplify the expression

Simplify the terms by performing the arithmetic operations:\[ (2r^{3} - 5r^{3}) + (3rs - rs) + (-5s^{2} - 4s^{2}) \]Compute each group:\[ 2r^{3} - 5r^{3} = -3r^{3} \]\[ 3rs - rs = 2rs \]\[ -5s^{2} - 4s^{2} = -9s^{2} \]Combine the results to get the final simplified expression:\[ -3r^{3} + 2rs - 9s^{2} \]

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Key Concepts

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

distributing negative sign

In polynomial subtraction, distributing the negative sign allows you to correctly subtract each term of the polynomial. Suppose you have two polynomials:

First polynomial: $2r^{3} + 3rs - 5s^{2}$
Second polynomial: $5r^{3} + rs + 4s^{2}$

When subtracting, you distribute the negative sign to each term in the second polynomial. This means that each term changes its sign:

Positive becomes negative
Negative becomes positive

After distribution, the second polynomial looks like this:

$-5r^{3}$
$-rs$
$-4s^{2}$

Now you can properly add or subtract these terms with those in the first polynomial.

combining like terms

Combining like terms is essential for simplifying polynomials. Like terms are terms that have the same variables and the same exponents. For example, in the expression:

$2r^{3}$
$-5r^{3}$
$3rs$
$-rs$
$-5s^{2}$
$-4s^{2}$

You need to group the like terms together:

$2r^{3}$ and $-5r^{3}$
$3rs$ and $-rs$
$-5s^{2}$ and $-4s^{2}$

By grouping them, it becomes easier to combine and simplify the expression further.

simplifying expressions

After distributing the negative sign and combining like terms, the next step is to perform the arithmetic operations. Simplify each group by adding or subtracting the coefficients of like terms:

$2r^{3} - 5r^{3} = -3r^{3}$
$3rs - rs = 2rs$
$-5s^{2} - 4s^{2} = -9s^{2}$

Combine the simplified terms to get the final expression:

\[-3r^{3} + 2rs - 9s^{2}\]

Following these steps ensures you correctly simplify polynomial expressions with subtraction, yielding a clear and concise result.

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91影视

$$ \left(2 r^{3}+3 r s-5 s^{2}\right)-\left(5 r^{3}+r s+4 s^{2}\right) $$

Short Answer

Step by step solution

- Distribute the negative sign

- Combine like terms

- Simplify the expression

Key Concepts

distributing negative sign

combining like terms

simplifying expressions

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