Problem 65 Multiplying Polynomials Multip... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 65

Multiplying Polynomials Multiply. $$(2 m-n)(2 m+n)-(m-2 n)^{2}$$

Short Answer

Expert verified

3m^2 + 4mn - 5n^2

Step by step solution

Expand the First Expression

Use the difference of squares formula for $ (2m - n)(2m + n) $ which is $ a^2 - b^2 $. Here, $ a = 2m $ and $ b = n $, so it becomes $ (2m)^2 - (n)^2 = 4m^2 - n^2 $.

Expand the Second Expression

Use the square of a binomial formula for $ (m - 2n)^2 $ which is $ a^2 - 2ab + b^2 $. Here, $ a = m $ and $ b = 2n $, so it becomes $ m^2 - 4mn + 4n^2 $.

Simplify the Entire Expression

Combine the results of the first two steps: $ (2m - n)(2m + n) - (m - 2n)^2 $ which translates to $ 4m^2 - n^2 - (m^2 - 4mn + 4n^2) $. Distribute the negative sign: $ 4m^2 - n^2 - m^2 + 4mn - 4n^2 $.

Collect Like Terms

Combine like terms: $ 4m^2 - m^2 $ and $ -n^2 - 4n^2 $ remain the same but added together appropriately: $ 3m^2 + 4mn - 5n^2 $.

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Key Concepts

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

Difference of Squares

The difference of squares formula is a key concept in algebra. It helps in simplifying expressions where two squares are subtracted. The general formula is given by: $a^2 - b^2 = (a - b)(a + b)$. Here, a and b can be any expressions.

This formula is very useful because it transforms a potentially complicated multiplication problem into an easy subtraction of squares.
In our exercise, we use the difference of squares formula like this: $(2m - n)(2m + n) = (2m)^2 - (n)^2$. This simplifies further: \[4m^2 - n^2\].

So, the difference of squares formula turns the product of two binomials into a simple subtraction.

Binomial Expansion

The binomial expansion formula helps in expanding expressions where a binomial is raised to a power. The formula for the square of a binomial $(a + b)^2$ is: $a^2 + 2ab + b^2$.

For our exercise, we use the formula to expand $(m - 2n)^2$: $(m - 2n)^2 = m^2 - 2 \times m \times 2n + (2n)^2$. This simplifies to: \[m^2 - 4mn + 4n^2\].

This expansion is crucial for simplifying expressions, as seen in the given exercise. It allows us to convert a compact form into a much more manageable expression.

Polynomial Simplification

Simplifying polynomials involves combining like terms to make the expression easier to handle. After expanding both parts of our polynomial, we need to put everything together.

From the exercise:

First part: $4m^2 - n^2$
Second part (with a negative sign): $-(m^2 - 4mn + 4n^2)$

Now, distributing the negative sign gives:

$4m^2 - n^2$
$-(m^2) + 4mn - 4n^2$

Combining like terms results in:

$4m^2 - m^2 = 3m^2$
$-n^2 - 4n^2 = -5n^2$
$4mn$ (only term)

Therefore, the final simplified form is: \[3m^2 + 4mn - 5n^2\].
Simplifying polynomials helps in making the expression concise and readable.

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91影视

Multiplying Polynomials Multiply. $$(2 m-n)(2 m+n)-(m-2 n)^{2}$$

Short Answer

Step by step solution

Expand the First Expression

Expand the Second Expression

Simplify the Entire Expression

Collect Like Terms

Key Concepts

Difference of Squares

Binomial Expansion

Polynomial Simplification

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