Identifying the Type of Special Product

Oct 13, 2025 by leolavanderia.com.br 86 views

Understanding the Problem

When we are given an expression like $$(-2m + 5n)^2$$, we need to determine its equivalent form and identify the type of special product it represents. In this case, we are looking for the expanded form of the given expression and the type of special product it is.

Expanding the Expression

To find the equivalent form of $$(-2m + 5n)^2$$, we need to apply the formula for squaring a binomial:

(a + b)^2 = a^2 + 2ab + b^2

In this case, $$a = -2m$$ and $$b = 5n$$. Substituting these values into the formula, we get:

( -2m + 5n )^2 = ( -2m )^2 + 2( -2m )( 5n ) + ( 5n )^2

Expanding each term, we get:

( -2m )^2 = 4m^2

2( -2m )( 5n ) = -20mn

( 5n )^2 = 25n^2

Combining these terms, we get:

4m^2 - 20mn + 25n^2

Identifying the Type of Special Product

Now that we have the expanded form of the expression, we need to identify the type of special product it represents. In this case, we can see that the expression is a perfect square trinomial, which is a special product that can be factored into the square of a binomial.

Perfect Square Trinomials

A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It has the form:

a^2 + 2ab + b^2 = (a + b)^2

In our case, the expression $$4m^2 - 20mn + 25n^2$$ is a perfect square trinomial because it can be factored into the square of a binomial:

( -2m + 5n )^2 = 4m^2 - 20mn + 25n^2

Conclusion

In conclusion, the equivalent form of $$(-2m + 5n)^2$$ is $$4m^2 - 20mn + 25n^2$$, and it is a perfect square trinomial. This type of special product can be factored into the square of a binomial, making it an important concept in algebra.

Key Takeaways

To expand an expression like $$(-2m + 5n)^2$$, we need to apply the formula for squaring a binomial.
The expanded form of the expression is $$4m^2 - 20mn + 25n^2$$.
The expression $$4m^2 - 20mn + 25n^2$$ is a perfect square trinomial because it can be factored into the square of a binomial.
Perfect square trinomials are an important type of special product in algebra, and they can be factored into the square of a binomial.

Examples and Exercises

Expand the expression $$(3x - 2y)^2$$.
Identify the type of special product represented by the expression $$9x^2 - 12xy + 4y^2$$.
Factor the expression $$4x^2 + 12xy + 9y^2$$ into the square of a binomial.

Answer Key

$(3x - 2y)^2 = 9x^2 - 12xy + 4y^2$
The expression $$9x^2 - 12xy + 4y^2$$ is a perfect square trinomial.
The expression $$4x^2 + 12xy + 9y^2$$ can be factored into the square of a binomial: $$(2x + 3y)^2$$
Frequently Asked Questions (FAQs) about Squaring Binomials and Special Products ====================================================================================

Q: What is a binomial?

A: A binomial is an algebraic expression consisting of two terms, such as $$a + b$$ or $$a - b$$.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It has the form:

a^2 + 2ab + b^2 = (a + b)^2

Q: How do I expand a binomial squared?

A: To expand a binomial squared, you can use the formula:

(a + b)^2 = a^2 + 2ab + b^2

Q: What is the difference of squares?

A: The difference of squares is a special product that can be factored into the product of two binomials. It has the form:

a^2 - b^2 = (a + b)(a - b)

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you can use the formula:

a^2 + 2ab + b^2 = (a + b)^2

Q: What is the product of two binomials?

A: The product of two binomials is a special product that can be factored into the product of two binomials. It has the form:

(a + b)(c + d) = ac + ad + bc + bd

Q: How do I expand a product of two binomials?

A: To expand a product of two binomials, you can use the distributive property:

(a + b)(c + d) = ac + ad + bc + bd

Q: What are some common special products in algebra?

A: Some common special products in algebra include:

Perfect square trinomials
Difference of squares
Product of two binomials
Sum of cubes
Difference of cubes

Q: How do I identify a special product in an algebraic expression?

A: To identify a special product in an algebraic expression, you can look for patterns such as:

A perfect square trinomial: $$a^2 + 2ab + b^2$$
A difference of squares: $$a^2 - b^2$$
A product of two binomials: $$(a + b)(c + d)$$

Q: What are some tips for working with special products in algebra?

A: Some tips for working with special products in algebra include:

Look for patterns in the expression
Use formulas and theorems to simplify the expression
Factor the expression into its simplest form
Check your work by plugging in values or using a calculator

Conclusion

In conclusion, squaring binomials and special products are important concepts in algebra that can be used to simplify and factor algebraic expressions. By understanding these concepts and practicing with examples, you can become proficient in working with special products and expand your algebraic skills.