Not verifying the identity

Oct 13, 2025 by miodeya.com 109 views

Introduction

In mathematics, identities are equations that are true for all possible values of the variables involved. They are used to simplify expressions, solve equations, and prove theorems. In this article, we will discuss five identities and determine which ones are true.

Identity A: $(x+2)^3 = x^3 + 8$

This identity is a claim that the cube of the binomial $(x+2)$ is equal to $x^3 + 8$ . To verify this, we can expand the left-hand side using the binomial theorem.

(x+2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3

Simplifying the expression, we get:

(x+2)^3 = x^3 + 6x^2 + 12x + 8

Comparing this with the right-hand side, we see that the two expressions are not equal. Therefore, Identity A is false.

Identity B: $(x^6 + x) = (x-1)(x^5 + x^4 + x^3 + x^2 + x)$

This identity is a claim that the sum of $x^6$ and $x$ is equal to the product of $(x-1)$ and the polynomial $x^5 + x^4 + x^3 + x^2 + x$ . To verify this, we can expand the right-hand side using the distributive property.

(x-1)(x^5 + x^4 + x^3 + x^2 + x) = x^6 + x^5 + x^4 + x^3 + x^2 + x - x^5 - x^4 - x^3 - x^2 - x

Simplifying the expression, we get:

(x-1)(x^5 + x^4 + x^3 + x^2 + x) = x^6 - x

Comparing this with the left-hand side, we see that the two expressions are not equal. Therefore, Identity B is false.

Identity C: $(x^2 - 1)(x^4 + x^2 + 1) = x^6 - 1$

This identity is a claim that the product of $(x^2 - 1)$ and $(x^4 + x^2 + 1)$ is equal to $x^6 - 1$ . To verify this, we can expand the left-hand side using the distributive property.

(x^2 - 1)(x^4 + x^2 + 1) = x^6 + x^4 - x^2 + x^2 - x^4 - x^2 + x^2 - 1

Simplifying the expression, we get:

(x^2 - 1)(x^4 + x^2 + 1) = x^6 - 1

Comparing this with the right-hand side, we see that the two expressions are equal. Therefore, Identity C is true.

Identity D: $(x+1)^4 = x^4 + x^3 + x^2 + x + 1$

This identity is a claim that the fourth power of $(x+1)$ is equal to $x^4 + x^3 + x^2 + x + 1$ . To verify this, we can expand the left-hand side using the binomial theorem.

(x+1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1

Comparing this with the right-hand side, we see that the two expressions are not equal. Therefore, Identity D is false.

Identity E: $(x+1)(x^4 - x^3 + x^2 - x + 1) = x^5 - x^4 + x^3 - x^2 + x$

This identity is a claim that the product of $(x+1)$ and $(x^4 - x^3 + x^2 - x + 1)$ is equal to $x^5 - x^4 + x^3 - x^2 + x$ . To verify this, we can expand the left-hand side using the distributive property.

(x+1)(x^4 - x^3 + x^2 - x + 1) = x^5 - x^4 + x^3 - x^2 + x + x^4 - x^3 + x^2 - x + 1

Simplifying the expression, we get:

(x+1)(x^4 - x^3 + x^2 - x + 1) = x^5 - x^4 + x^3 - x^2 + x + 1

Comparing this with the right-hand side, we see that the two expressions are equal. Therefore, Identity E is true.

Conclusion

Q: What is an identity in mathematics?

A: An identity in mathematics is an equation that is true for all possible values of the variables involved. Identities are used to simplify expressions, solve equations, and prove theorems.

Q: How do I verify an identity?

A: To verify an identity, you can use various methods such as:

Expanding the left-hand side using the distributive property or the binomial theorem
Simplifying the expression by combining like terms
Comparing the resulting expression with the right-hand side

Q: What are some common types of identities?

A: Some common types of identities include:

Algebraic identities: These are identities that involve algebraic expressions, such as $(x+y)^2 = x^2 + 2xy + y^2$
Trigonometric identities: These are identities that involve trigonometric functions, such as $\sin^2(x) + \cos^2(x) = 1$
Exponential identities: These are identities that involve exponential functions, such as $e^x \cdot e^y = e^{x+y}$

Q: How do I use identities to simplify expressions?

A: To use identities to simplify expressions, you can:

Expand the expression using the distributive property or the binomial theorem
Simplify the resulting expression by combining like terms
Use algebraic manipulations to rewrite the expression in a simpler form

Q: Can I use identities to solve equations?

A: Yes, you can use identities to solve equations. For example, if you have an equation of the form $x^2 + 2x + 1 = 0$ , you can use the identity $(x+1)^2 = x^2 + 2x + 1$ to rewrite the equation as $(x+1)^2 = 0$ . This allows you to solve for $x$ .

Q: What are some common mistakes to avoid when working with identities?

A: Some common mistakes to avoid when working with identities include:

Not verifying the identity: Make sure to verify the identity before using it to simplify expressions or solve equations.
Not simplifying the expression: Make sure to simplify the expression by combining like terms and using algebraic manipulations.
Not checking for extraneous solutions: Make sure to check for extraneous solutions when solving equations using identities.

Q: How do I know if an identity is true or false?

A: To determine if an identity is true or false, you can:

Verify the identity by expanding the left-hand side and comparing it with the right-hand side
Use algebraic manipulations to simplify the expression and check if it is equal to the right-hand side
Check if the identity is a known algebraic identity or a well-known mathematical fact.

Conclusion

In this article, we have discussed frequently asked questions about identities and algebra. We have covered topics such as verifying identities, simplifying expressions, solving equations, and common mistakes to avoid. By following these tips and techniques, you can become more confident and proficient in working with identities and algebra.