Problem 64 Add or subtract the polynomials.... [FREE SOLUTION]

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Chapter 6: Problem 64

Add or subtract the polynomials. $$\left(m^{2}+2 n^{2}\right)-\left(m^{2}-8 m n-n^{2}\right)$$

Short Answer

Expert verified

3n^2 + 8mn

Step by step solution

Write the Polynomials

First, write down the given polynomials: $ (m^2 + 2n^2) - (m^2 - 8mn - n^2) $

Distribute the Negative Sign

Distribute the negative sign to each term inside the parentheses: $ m^2 + 2n^2 - m^2 + 8mn + n^2 $

Combine Like Terms

Combine the like terms: $ (m^2 - m^2) + (2n^2 + n^2) + 8mn $Simplify the expression: $ 0 + 3n^2 + 8mn = 3n^2 + 8mn $

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Key Concepts

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

Adding Polynomials

Adding polynomials is quite easy once you understand the process. Each polynomial is composed of terms. To add polynomials, you simply add their corresponding terms. Make sure that you are combining terms that have the same variables raised to the same power. These are called 'like terms'.

For example, to add $3x^2 + 5x + 2$ and $4x^2 - 2x + 7$:

First, identify and align the like terms: $3x^2$ and $4x^2$, $5x$ and $-2x$, and the constants $2$ and $7$.

Next, add the coefficients of like terms: $3x^2 + 4x^2$, $5x - 2x$, and $2 + 7$.

The resulting polynomial is $7x^2 + 3x + 9$.

This process remains the same no matter how many terms the polynomials have. Make sure you correctly align and add the like terms for accurate results.

Subtracting Polynomials

Subtracting polynomials is almost identical to adding them, but with an extra step. When subtracting, you need to distribute the negative sign to each term in the polynomial being subtracted. This changes the sign of every term in that polynomial.

For example, to subtract $4x^2 - 2x + 7$ from $3x^2 + 5x + 2$:

First, write the subtraction as an addition of a negative: $(3x^2 + 5x + 2) - (4x^2 - 2x + 7)$.

Next, distribute the negative sign through the second polynomial: $(3x^2 + 5x + 2) + (-4x^2 + 2x - 7)$.

Now, combine like terms: $3x^2 - 4x^2$, $5x + 2x$, and $2 - 7$.

The result is $-x^2 + 7x - 5$.

Always remember to distribute the negative sign to each term of the polynomial you are subtracting. This is a critical step and missing it can lead to incorrect results.

Combining Like Terms

Combining like terms is a fundamental skill when working with polynomials. Like terms are those that have the same variables raised to the same powers. Only like terms can be combined. When combining, you add or subtract their coefficients, and the variable part remains unchanged.

For example, in the expression $3x^2 + 4x - 5 + 2x^2 - 3x + 7$, identify like terms: $3x^2$ and $2x^2$, $4x$ and $-3x$, and the constants $-5$ and $7$.

Combine the like terms: $3x^2 + 2x^2 = 5x^2$, $4x - 3x = x$, and $-5 + 7 = 2$.

The simplified expression is $5x^2 + x + 2$.

This process is essential when adding or subtracting polynomials since you'll often need to combine like terms as the final step. Always take care to combine only those terms that are exactly alike in both variable and exponent.

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91影视

Add or subtract the polynomials. $$\left(m^{2}+2 n^{2}\right)-\left(m^{2}-8 m n-n^{2}\right)$$

Short Answer

Step by step solution

Write the Polynomials

Distribute the Negative Sign

Combine Like Terms

Key Concepts

Adding Polynomials

Subtracting Polynomials

Combining Like Terms

One App. One Place for Learning.

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