Simplify The Expression: { (x-2)\left(x^2+x-1\right)$}$

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Introduction

In this article, we will simplify the given expression (x−2)(x2+x−1)(x-2)\left(x^2+x-1\right). This involves multiplying the two binomials together using the distributive property. We will also explore the concept of like terms and how to combine them to simplify the expression.

Understanding the Expression

The given expression is a product of two binomials: (x−2)(x-2) and (x2+x−1)\left(x^2+x-1\right). To simplify this expression, we need to multiply each term in the first binomial by each term in the second binomial.

Multiplying the Binomials

To multiply the binomials, we will use the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. We will apply this property to each term in the first binomial.

Step 1: Multiply the First Term in the First Binomial by Each Term in the Second Binomial

The first term in the first binomial is xx. We will multiply this term by each term in the second binomial: x2x^2, xx, and −1-1.

x⋅x2=x3x \cdot x^2 = x^3 x⋅x=x2x \cdot x = x^2 x⋅−1=−xx \cdot -1 = -x

Step 2: Multiply the Second Term in the First Binomial by Each Term in the Second Binomial

The second term in the first binomial is −2-2. We will multiply this term by each term in the second binomial: x2x^2, xx, and −1-1.

−2⋅x2=−2x2-2 \cdot x^2 = -2x^2 −2⋅x=−2x-2 \cdot x = -2x −2⋅−1=2-2 \cdot -1 = 2

Combining Like Terms

Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

Combining the Terms with x3x^3

There is only one term with x3x^3, which is x3x^3.

Combining the Terms with x2x^2

There are two terms with x2x^2: x2x^2 and −2x2-2x^2. We can combine these terms by adding their coefficients.

x2−2x2=−x2x^2 - 2x^2 = -x^2

Combining the Terms with xx

There are two terms with xx: x2x^2 is not a term with x, and xx and −2x-2x. We can combine these terms by adding their coefficients.

x−2x=−xx - 2x = -x

Combining the Constant Terms

There is only one constant term, which is 22.

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression by adding the remaining terms.

x3−x2−x+2x^3 - x^2 - x + 2

Conclusion

In this article, we simplified the expression (x−2)(x2+x−1)(x-2)\left(x^2+x-1\right) by multiplying the two binomials together using the distributive property and combining like terms. The simplified expression is x3−x2−x+2x^3 - x^2 - x + 2.

Final Answer

The final answer is x3−x2−x+2\boxed{x^3 - x^2 - x + 2}.

Related Topics

  • Distributive Property
  • Like Terms
  • Combining Like Terms
  • Simplifying Expressions

References

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Introduction

In our previous article, we simplified the expression (x−2)(x2+x−1)(x-2)\left(x^2+x-1\right) by multiplying the two binomials together using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. This means that we can multiply a single term by two or more terms inside a set of parentheses.

Q: How do I simplify an expression with multiple binomials?

A: To simplify an expression with multiple binomials, we need to multiply each term in the first binomial by each term in the second binomial, and then combine like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, x2x^2 and −x2-x^2 are like terms because they both have the variable xx raised to the power of 2.

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract the coefficients of the like terms. For example, if we have the terms x2x^2 and −x2-x^2, we can combine them by adding their coefficients: x2−x2=0x^2 - x^2 = 0.

Q: What is the difference between a term and a factor?

A: A term is a single expression that consists of a variable or a constant, such as xx or −2-2. A factor is a term that is multiplied by another term, such as x⋅2x \cdot 2.

Q: How do I simplify an expression with parentheses?

A: To simplify an expression with parentheses, we need to follow the order of operations (PEMDAS):

  1. Evaluate any expressions inside the parentheses.
  2. Multiply any terms that are multiplied together.
  3. Add or subtract any terms that are added or subtracted together.

Q: What is the final answer to the expression (x−2)(x2+x−1)(x-2)\left(x^2+x-1\right)?

A: The final answer to the expression (x−2)(x2+x−1)(x-2)\left(x^2+x-1\right) is x3−x2−x+2x^3 - x^2 - x + 2.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions. We covered topics such as the distributive property, like terms, combining like terms, and simplifying expressions with parentheses.

Final Answer

The final answer is x3−x2−x+2\boxed{x^3 - x^2 - x + 2}.

Related Topics

  • Distributive Property
  • Like Terms
  • Combining Like Terms
  • Simplifying Expressions
  • Order of Operations (PEMDAS)

References