Problem 87 $$\text { Subtract: }\left(2 x^{... [FREE SOLUTION]

Chapter 1: Problem 87

$$\text { Subtract: }\left(2 x^{2}+3 x-4\right)-\left(x^{2}+3 x-5\right)[\text { A.2 }]$$

Short Answer

Expert verified

The result of the subtraction $ (2x^{2} + 3x - 4) - (x^{2} + 3x - 5) $ is $x^{2} + 1 $

Step by step solution

Distribute the Negative Sign

Distribute the negative sign through the second set of parentheses, modifying the sign of each term within. Therefore, $ (x^{2}+3x-5) $ changes to $ -(x^{2})-3x+5 $

Combine Like Terms

Add the like terms (the terms in both polynomials that contain the same variable raised to the same power), namely $x^{2}$, $x$, and the constants. The new expression will be $ (2x^{2}-(x^{2}) + 3x - 3x - 4 + 5 ) $

Simplify the Expression

Simplify by carrying out the operation in step 2. The simplified expression is $x^{2}+1$

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Key Concepts

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

Distributing Negative Signs

When subtracting polynomials, one important step is distributing the negative sign.
You can think of a negative sign as a multiplier of (-1) for everything within the parentheses. Let's break down what this means.

Take the polynomial being subtracted: $(x^{2} + 3x - 5)$.
Change the sign of each term: $x^{2}$ becomes $-x^{2}$, $3x$ becomes $-3x$, and $-5$ becomes $+5$.

By distributing the negative sign, we transform the entire polynomial into its additive inverse.
Now we have $2x^{2} + 3x - 4 + [-x^{2} - 3x + 5]$.
This prepares you for combining like terms, ensuring you correctly incorporate the sign changes.

Combining Like Terms

The next step in subtracting polynomials involves combining like terms.
This means looking for terms that have the same variable and the same power.

First we have $2x^{2}$ and $-x^{2}$. Both are $x^{2}$ terms.
We also have $3x$ and $-3x$; these are both $x$ terms.
Lastly, compare constants: we have $-4$ and $+5$.

Group and add or subtract these like terms.
The $x^{2}$ terms combine to $(2x^{2} - x^{2}) = x^{2}$.
The $x$ terms cancel each other out, since $3x - 3x = 0$.
The constants combine to $(-4 + 5) = 1$.
This results in the expression $(x^{2} + 1)$.

Simplifying Expressions

Once like terms are combined, the final step is to simplify the overall expression. Simplification means writing the expression in its simplest form,
without any unnecessary terms or redundancy.
This expression, $x^{2} + 1$, is already in its simplest form.

All like terms have been combined effectively.
The operations have been performed accurately, with no further simplification possible.

In practice, simplifying expressions ensures clarity and efficiency in mathematical work.
It reveals the true nature of the solution, free from clutter and complexity.
Such simplification is crucial for solving equations, modeling, and understanding the actual relationships between variables.

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91影视

$$\text { Subtract: }\left(2 x^{2}+3 x-4\right)-\left(x^{2}+3 x-5\right)[\text { A.2 }]$$

Short Answer

Step by step solution

Distribute the Negative Sign

Combine Like Terms

Simplify the Expression

Key Concepts

Distributing Negative Signs

Combining Like Terms

Simplifying Expressions

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