Write The Following Expression As A Simplified Polynomial In Standard Form. { (x-4)^2 + 4(x-4) + 7$}$

Introduction

Polynomials are a fundamental concept in algebra, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore how to simplify the given expression $(x-4)^2 + 4(x-4) + 7$ and express it as a simplified polynomial in standard form.

Understanding the Expression

The given expression is a quadratic expression, which means it contains a squared variable. The expression is $(x-4)^2 + 4(x-4) + 7$. To simplify this expression, we need to expand the squared term and combine like terms.

Expanding the Squared Term

The first step in simplifying the expression is to expand the squared term $(x-4)^2$. To do this, we need to use the formula $(a-b)^2 = a^2 - 2ab + b^2$. In this case, $a = x$ and $b = 4$. Therefore, we can expand the squared term as follows:

$(x-4)^2 = x^2 - 2(x)(4) + 4^2$ $(x-4)^2 = x^2 - 8x + 16$

Expanding the Expression

Now that we have expanded the squared term, we can rewrite the original expression as follows:

$(x-4)^2 + 4(x-4) + 7$ $x^2 - 8x + 16 + 4(x-4) + 7$

Combining Like Terms

The next step in simplifying the expression is to combine like terms. To do this, we need to identify the like terms in the expression. In this case, the like terms are the terms that contain the variable $x$. The like terms are $-8x$ and $4(x-4)$.

To combine these like terms, we need to multiply the coefficient of the second term by the value inside the parentheses. In this case, the coefficient of the second term is $4$, and the value inside the parentheses is $(x-4)$. Therefore, we can multiply the coefficient by the value inside the parentheses as follows:

$4(x-4) = 4x - 16$

Now that we have multiplied the coefficient by the value inside the parentheses, we can rewrite the expression as follows:

$x^2 - 8x + 16 + 4x - 16 + 7$

Simplifying the Expression

The final step in simplifying the expression is to combine the like terms. To do this, we need to add the coefficients of the like terms. In this case, the like terms are $-8x$ and $4x$. The coefficient of the first term is $-8$, and the coefficient of the second term is $4$. Therefore, we can add the coefficients as follows:

$-8x + 4x = -4x$

Now that we have added the coefficients, we can rewrite the expression as follows:

$x^2 - 4x + 16 - 16 + 7$

Final Simplification

The final step in simplifying the expression is to combine the constants. In this case, the constants are $16$ and $-16$. The sum of these constants is $0$. Therefore, we can rewrite the expression as follows:

$x^2 - 4x + 7$

Conclusion

In this article, we have simplified the given expression $(x-4)^2 + 4(x-4) + 7$ and expressed it as a simplified polynomial in standard form. We have expanded the squared term, combined like terms, and simplified the expression. The final simplified expression is $x^2 - 4x + 7$.

Key Takeaways

• To simplify a polynomial, we need to expand the squared terms and combine like terms.
• To combine like terms, we need to add the coefficients of the like terms.
• To simplify the expression, we need to combine the constants.

Practice Problems

1. Simplify the expression $(x+3)^2 + 2(x+3) + 5$.
2. Simplify the expression $(x-2)^2 + 3(x-2) + 9$.
3. Simplify the expression $(x+1)^2 + 4(x+1) + 11$.

Answer Key

1. $x^2 + 8x + 16 + 6x + 18 + 5 = x^2 + 14x + 39$
2. $x^2 - 4x + 4 + 3x - 6 + 9 = x^2 - x + 7$
3. $x^2 + 2x + 1 + 4x + 4 + 11 = x^2 + 6x + 16$
   Simplifying Polynomials: A Q&A Guide =====================================

Introduction

In our previous article, we explored how to simplify the given expression $(x-4)^2 + 4(x-4) + 7$ and express it as a simplified polynomial in standard form. In this article, we will answer some frequently asked questions about simplifying polynomials.

Q: What is a polynomial?

A polynomial is an expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified into different types, such as linear, quadratic, cubic, and so on, based on the degree of the polynomial.

Q: What is the degree of a polynomial?

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial $x^2 + 3x + 2$, the degree is 2 because the highest power of the variable $x$ is 2.

Q: How do I simplify a polynomial?

To simplify a polynomial, you need to follow these steps:

1. Expand the squared terms using the formula $(a-b)^2 = a^2 - 2ab + b^2$.
2. Combine like terms by adding the coefficients of the like terms.
3. Simplify the expression by combining the constants.

Q: What are like terms?

Like terms are terms that contain the same variable raised to the same power. For example, in the polynomial $x^2 + 3x + 2$, the terms $x^2$ and $3x$ are like terms because they both contain the variable $x$ raised to the power of 2.

Q: How do I combine like terms?

To combine like terms, you need to add the coefficients of the like terms. For example, in the polynomial $x^2 + 3x + 2$, the like terms are $x^2$ and $3x$. To combine these like terms, you need to add the coefficients, which gives you $x^2 + 4x + 2$.

Q: What is the standard form of a polynomial?

The standard form of a polynomial is the form in which the terms are arranged in descending order of the powers of the variable. For example, in the polynomial $x^2 + 3x + 2$, the standard form is $x^2 + 3x + 2$.

Q: How do I simplify a polynomial with multiple variables?

To simplify a polynomial with multiple variables, you need to follow the same steps as before:

1. Expand the squared terms using the formula $(a-b)^2 = a^2 - 2ab + b^2$.
2. Combine like terms by adding the coefficients of the like terms.
3. Simplify the expression by combining the constants.

However, when simplifying a polynomial with multiple variables, you need to be careful to keep track of the different variables and their powers.

Q: What are some common mistakes to avoid when simplifying polynomials?

Some common mistakes to avoid when simplifying polynomials include:

• Not expanding the squared terms correctly
• Not combining like terms correctly
• Not simplifying the expression correctly
• Not keeping track of the different variables and their powers

Conclusion

In this article, we have answered some frequently asked questions about simplifying polynomials. We have covered topics such as the definition of a polynomial, the degree of a polynomial, and how to simplify a polynomial. We have also discussed some common mistakes to avoid when simplifying polynomials.

Practice Problems

1. Simplify the expression $(x+3)^2 + 2(x+3) + 5$.
2. Simplify the expression $(x-2)^2 + 3(x-2) + 9$.
3. Simplify the expression $(x+1)^2 + 4(x+1) + 11$.

Answer Key

1. $x^2 + 8x + 16 + 6x + 18 + 5 = x^2 + 14x + 39$
2. $x^2 - 4x + 4 + 3x - 6 + 9 = x^2 - x + 7$
3. $x^2 + 2x + 1 + 4x + 4 + 11 = x^2 + 6x + 16$