Problem 41 Add or subtract the polynomials.... [FREE SOLUTION]

Chapter 6: Problem 41

Add or subtract the polynomials. $$\left(4 m^{2}-6 m-3\right)-\left(2 m^{2}+m-7\right)$$

Short Answer

Expert verified

$2m^2 - 7m + 4$

Step by step solution

Distribute the Negative Sign

Convert the subtraction of the second polynomial into the addition of its opposite. Distribute the negative sign to each term in the parentheses: \[(4m^2 - 6m - 3) - (2m^2 + m - 7) = 4m^2 - 6m - 3 - 2m^2 - m + 7\]

Group Like Terms

Group together the like terms to make it easier to combine them: \[(4m^2 - 2m^2) + (-6m - m) + (-3 + 7)\]

Combine Like Terms

Combine the coefficients of like terms to simplify the expression: \[ 4m^2 - 2m^2 = 2m^2 \] \[( -6m - m = -7m) \] \[( -3 + 7 = 4) \] So the result is: \[(2m^2 - 7m + 4)\]

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Key Concepts

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

Adding Polynomials

Adding polynomials involves combining corresponding terms from each polynomial. Each term in a polynomial is made up of a coefficient and a variable raised to an exponent. Ensure that when you add polynomials, you only combine the like terms. These are terms that have the same variable raised to the same power.

Here is a quick guide to adding polynomials:

Identify the like terms in both polynomials. These will typically be terms that have the same variable parts.
Combine the coefficients of like terms. This means adding the numbers in front of the variables together.
Arrange the result in standard form. This means writing the terms in descending order of their exponents.

As an example, consider adding $3x^2 + 5x + 2$ and $2x^2 + 3x - 4$:
\[ (3x^2 + 2x^2) + (5x + 3x) + (2 - 4) = 5x^2 + 8x - 2 \]
In this example, like terms are added to get the resulting polynomial.

Subtracting Polynomials

Subtracting polynomials is similar to adding them, but it involves an extra step: distributing the negative sign across the second polynomial before you combine the like terms.
Remember these steps for subtracting polynomials:

First, rewrite the subtraction problem as adding the opposite. This means distributing the negative sign to each term of the polynomial being subtracted.
Then, identify and combine like terms by adding the coefficients.
Finally, write the simplified polynomial in standard form.

As an example, our original exercise involves subtracting $2m^2 + m - 7$ from $4m^2 - 6m - 3$:
1. Distribute the negative sign:

\[ 4m^2 - 6m - 3 - 2m^2 - m + 7 \]

2. Group and combine like terms:

\[ (4m^2 - 2m^2) + (-6m - m) + (-3 + 7) = 2m^2 - 7m + 4 \]
3. The final simplified polynomial is $2m^2 - 7m + 4$

Combining Like Terms

Combining like terms is a fundamental skill in simplifying polynomials. This process involves gathering and then adding or subtracting the coefficients of terms that have identical variable parts.
Here's a step-by-step method on how to combine like terms:

Identify like terms: These are terms that have the same variables raised to the same power.
Group the like terms together for easier calculation.
Sum up or subtract the coefficients of these like terms.

As seen in our exercise, we started with:

\[ 4m^2 - 6m - 3 - 2m^2 - m + 7 \]

Our like terms are:

$4m^2 - 2m^2$
$-6m - m$
$-3 + 7$

When combined, they give:

\[2m^2 - 7m + 4 \]
Combining like terms simplifies polynomials and makes further operations, such as addition or subtraction, much more manageable.

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91影视

Add or subtract the polynomials. $$\left(4 m^{2}-6 m-3\right)-\left(2 m^{2}+m-7\right)$$

Short Answer

Step by step solution

Distribute the Negative Sign

Group Like Terms

Combine Like Terms

Key Concepts

Adding Polynomials

Subtracting Polynomials

Combining Like Terms

One App. One Place for Learning.

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