Expand The Expression To A Polynomial In Standard Form:$\left(2x^2 + X + 2\right)\left(x^2 - 7x - 6\right$\]Answer:$\square$

Introduction

In algebra, expanding an expression to a polynomial in standard form is a crucial step in solving equations and manipulating mathematical expressions. In this article, we will focus on expanding the given expression $\left(2x^2 + x + 2\right)\left(x^2 - 7x - 6\right)$ to a polynomial in standard form.

Understanding the Expression

Before we begin expanding the expression, let's understand what it means to expand an expression to a polynomial in standard form. A polynomial in standard form is a mathematical expression consisting of variables and coefficients, where the variables are raised to non-negative integer powers. The standard form of a polynomial is typically written with the terms arranged in descending order of the powers of the variables.

Step 1: Multiply the Two Binomials

To expand the given expression, we need to multiply the two binomials $\left(2x^2 + x + 2\right)$ and $\left(x^2 - 7x - 6\right)$. We can do this by using the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

We will multiply each term in the first binomial by each term in the second binomial and then combine like terms.

Multiplying the Terms

Let's multiply the terms:

• $2x^2 \cdot x^2 = 2x^4$
• $2x^2 \cdot (-7x) = -14x^3$
• $2x^2 \cdot (-6) = -12x^2$
• $x \cdot x^2 = x^3$
• $x \cdot (-7x) = -7x^2$
• $x \cdot (-6) = -6x$
• $2 \cdot x^2 = 2x^2$
• $2 \cdot (-7x) = -14x$
• $2 \cdot (-6) = -12$

Combining Like Terms

Now, let's combine like terms:

• $2x^4$
• $-14x^3 + x^3 = -13x^3$
• $-12x^2 - 7x^2 + 2x^2 = -17x^2$
• $-6x - 14x = -20x$
• $-12$

The Expanded Expression

The expanded expression is:

$2x^4 - 13x^3 - 17x^2 - 20x - 12$

Conclusion

In this article, we expanded the given expression $\left(2x^2 + x + 2\right)\left(x^2 - 7x - 6\right)$ to a polynomial in standard form. We used the distributive property to multiply the two binomials and then combined like terms to simplify the expression. The expanded expression is $2x^4 - 13x^3 - 17x^2 - 20x - 12$.

Importance of Expanding Expressions

Expanding expressions to polynomials in standard form is an essential skill in algebra. It allows us to simplify complex expressions, solve equations, and manipulate mathematical expressions. In this article, we demonstrated how to expand a given expression using the distributive property and combining like terms.

Real-World Applications

Expanding expressions has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, expanding expressions is used to design and analyze complex systems, such as electrical circuits and mechanical systems. In physics, expanding expressions is used to describe the motion of objects and the behavior of physical systems.

Tips and Tricks

Here are some tips and tricks for expanding expressions:

• Use the distributive property to multiply binomials.
• Combine like terms to simplify the expression.
• Use the FOIL method to multiply two binomials.
• Use the binomial theorem to expand expressions with multiple terms.

Common Mistakes

Here are some common mistakes to avoid when expanding expressions:

• Failing to use the distributive property.
• Failing to combine like terms.
• Using the wrong method to multiply binomials.
• Making errors when simplifying the expression.

Conclusion

Introduction

In our previous article, we discussed how to expand expressions to polynomials in standard form. In this article, we will provide a Q&A guide to help you better understand the concept of expanding expressions and how to apply it in different situations.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to multiply a single term by a sum of terms.

Q: How do I apply the distributive property to expand expressions?

A: To apply the distributive property, you need to multiply each term in the first binomial by each term in the second binomial and then combine like terms. For example, to expand the expression $(2x^2 + x + 2)(x^2 - 7x - 6)$, you would multiply each term in the first binomial by each term in the second binomial and then combine like terms.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. FOIL stands for First, Outer, Inner, Last, which refers to the order in which you multiply the terms. The FOIL method is a shortcut for multiplying two binomials and is often used when the binomials have two terms each.

Q: How do I use the FOIL method to expand expressions?

A: To use the FOIL method, you need to multiply the first terms in each binomial, then the outer terms, then the inner terms, and finally the last terms. For example, to expand the expression $(x + 3)(x + 5)$, you would multiply the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: What is the binomial theorem?

A: The binomial theorem is a mathematical concept that describes the expansion of a binomial raised to a power. The binomial theorem is often used to expand expressions with multiple terms.

Q: How do I use the binomial theorem to expand expressions?

A: To use the binomial theorem, you need to identify the binomial and the power to which it is raised. You then need to apply the formula for the binomial theorem, which is $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.

Q: What are some common mistakes to avoid when expanding expressions?

A: Some common mistakes to avoid when expanding expressions include:

• Failing to use the distributive property.
• Failing to combine like terms.
• Using the wrong method to multiply binomials.
• Making errors when simplifying the expression.

Q: How do I simplify expressions after expanding them?

A: To simplify expressions after expanding them, you need to combine like terms and eliminate any unnecessary terms. You can use the distributive property to simplify expressions and the FOIL method to multiply binomials.

Q: What are some real-world applications of expanding expressions?

A: Expanding expressions has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, expanding expressions is used to design and analyze complex systems, such as electrical circuits and mechanical systems. In physics, expanding expressions is used to describe the motion of objects and the behavior of physical systems.

Conclusion

In conclusion, expanding expressions is an essential skill in algebra that has numerous real-world applications. In this article, we provided a Q&A guide to help you better understand the concept of expanding expressions and how to apply it in different situations. We also discussed common mistakes to avoid and how to simplify expressions after expanding them.