Select The Correct Answer.Two Buildings On Opposite Sides Of A Highway Are 3 X 3 − X 2 + 7 X + 100 3x^3 - X^2 + 7x + 100 3 X 3 − X 2 + 7 X + 100 Feet Apart. One Building Is 2 X 2 + 7 X 2x^2 + 7x 2 X 2 + 7 X Feet From The Highway. The Other Building Is X 3 + 2 X 2 − 18 X^3 + 2x^2 - 18 X 3 + 2 X 2 − 18 Feet From The Highway.

Problem Description

We are given two buildings on opposite sides of a highway. The distance between the two buildings is represented by the polynomial $3x^3 - x^2 + 7x + 100$. One building is $2x^2 + 7x$ feet from the highway, and the other building is $x^3 + 2x^2 - 18$ feet from the highway. Our goal is to find the correct answer by determining the distance between the two buildings.

Understanding the Problem

To solve this problem, we need to find the distance between the two buildings, which is represented by the polynomial $3x^3 - x^2 + 7x + 100$. We are also given the distances of the two buildings from the highway, which are $2x^2 + 7x$ and $x^3 + 2x^2 - 18$ feet, respectively.

Finding the Distance Between the Two Buildings

To find the distance between the two buildings, we need to add the distances of the two buildings from the highway. This can be represented by the equation:

$$\text{Distance} = (2x^2 + 7x) + (x^3 + 2x^2 - 18)$$

Simplifying the Equation

We can simplify the equation by combining like terms:

$$\text{Distance} = x^3 + 4x^2 + 7x - 18$$

Evaluating the Distance

To evaluate the distance, we need to find the value of the polynomial $x^3 + 4x^2 + 7x - 18$.

Using Algebraic Manipulation

We can use algebraic manipulation to simplify the polynomial and find its value. Let's start by factoring out the greatest common factor (GCF) of the polynomial:

$$x^3 + 4x^2 + 7x - 18 = x(x^2 + 4x + 7) - 18$$

Factoring the Quadratic Expression

We can factor the quadratic expression $x^2 + 4x + 7$:

$$x^2 + 4x + 7 = (x + 2)^2 + 3$$

Substituting the Factored Expression

We can substitute the factored expression back into the polynomial:

$$x(x^2 + 4x + 7) - 18 = x((x + 2)^2 + 3) - 18$$

Simplifying the Expression

We can simplify the expression by distributing the $x$ term:

$$x((x + 2)^2 + 3) - 18 = x(x^2 + 4x + 7) - 18$$

Evaluating the Expression

We can evaluate the expression by substituting the value of $x$:

$$x(x^2 + 4x + 7) - 18 = x(3x^2 + 7x - 18)$$

Finding the Value of the Polynomial

We can find the value of the polynomial by substituting the value of $x$:

$$x(3x^2 + 7x - 18) = 3x^3 + 7x^2 - 18x$$

Evaluating the Final Expression

We can evaluate the final expression by substituting the value of $x$:

$$3x^3 + 7x^2 - 18x = 3(3)^3 + 7(3)^2 - 18(3)$$

Simplifying the Final Expression

We can simplify the final expression by evaluating the powers of $3$:

$$3(3)^3 + 7(3)^2 - 18(3) = 3(27) + 7(9) - 54$$

Evaluating the Final Expression

We can evaluate the final expression by multiplying the numbers:

$$3(27) + 7(9) - 54 = 81 + 63 - 54$$

Simplifying the Final Expression

We can simplify the final expression by adding and subtracting the numbers:

$$81 + 63 - 54 = 90$$

Conclusion

The distance between the two buildings is $90$ feet.

Discussion

The problem requires us to find the distance between two buildings, which is represented by the polynomial $3x^3 - x^2 + 7x + 100$. We are also given the distances of the two buildings from the highway, which are $2x^2 + 7x$ and $x^3 + 2x^2 - 18$ feet, respectively. To solve this problem, we need to add the distances of the two buildings from the highway and simplify the resulting expression. We can use algebraic manipulation to simplify the polynomial and find its value. The final answer is $90$ feet.

Key Takeaways

• The distance between the two buildings is represented by the polynomial $3x^3 - x^2 + 7x + 100$.
• The distances of the two buildings from the highway are $2x^2 + 7x$ and $x^3 + 2x^2 - 18$ feet, respectively.
• To find the distance between the two buildings, we need to add the distances of the two buildings from the highway and simplify the resulting expression.
• We can use algebraic manipulation to simplify the polynomial and find its value.
• The final answer is $90$ feet.

Recommendations

• To solve this problem, you need to have a good understanding of algebraic manipulation and polynomial expressions.
• You should be able to add and simplify polynomial expressions.
• You should be able to use algebraic manipulation to simplify the polynomial and find its value.
• You should be able to evaluate the final expression and find the distance between the two buildings.

Conclusion

The problem requires us to find the distance between two buildings, which is represented by the polynomial $3x^3 - x^2 + 7x + 100$. We are also given the distances of the two buildings from the highway, which are $2x^2 + 7x$ and $x^3 + 2x^2 - 18$ feet, respectively. To solve this problem, we need to add the distances of the two buildings from the highway and simplify the resulting expression. We can use algebraic manipulation to simplify the polynomial and find its value. The final answer is $90$ feet.

Q: What is the problem asking for?

A: The problem is asking for the distance between two buildings, which is represented by the polynomial $3x^3 - x^2 + 7x + 100$. We are also given the distances of the two buildings from the highway, which are $2x^2 + 7x$ and $x^3 + 2x^2 - 18$ feet, respectively.

Q: How do we find the distance between the two buildings?

A: To find the distance between the two buildings, we need to add the distances of the two buildings from the highway and simplify the resulting expression.

Q: What is the first step in solving the problem?

A: The first step in solving the problem is to add the distances of the two buildings from the highway.

Q: How do we add the distances of the two buildings from the highway?

A: We can add the distances of the two buildings from the highway by combining like terms.

Q: What is the resulting expression after adding the distances of the two buildings from the highway?

A: The resulting expression after adding the distances of the two buildings from the highway is $x^3 + 4x^2 + 7x - 18$.

Q: How do we simplify the resulting expression?

A: We can simplify the resulting expression by factoring out the greatest common factor (GCF) of the polynomial.

Q: What is the GCF of the polynomial?

A: The GCF of the polynomial is $x$.

Q: How do we factor out the GCF of the polynomial?

A: We can factor out the GCF of the polynomial by dividing each term by the GCF.

Q: What is the resulting expression after factoring out the GCF of the polynomial?

A: The resulting expression after factoring out the GCF of the polynomial is $x(x^2 + 4x + 7) - 18$.

Q: How do we simplify the resulting expression further?

A: We can simplify the resulting expression further by factoring the quadratic expression $x^2 + 4x + 7$.

Q: What is the factored form of the quadratic expression $x^2 + 4x + 7$?

A: The factored form of the quadratic expression $x^2 + 4x + 7$ is $(x + 2)^2 + 3$.

Q: How do we substitute the factored form of the quadratic expression back into the polynomial?

A: We can substitute the factored form of the quadratic expression back into the polynomial by replacing $x^2 + 4x + 7$ with $(x + 2)^2 + 3$.

Q: What is the resulting expression after substituting the factored form of the quadratic expression back into the polynomial?

A: The resulting expression after substituting the factored form of the quadratic expression back into the polynomial is $x((x + 2)^2 + 3) - 18$.

Q: How do we simplify the resulting expression further?

A: We can simplify the resulting expression further by distributing the $x$ term.

Q: What is the resulting expression after distributing the $x$ term?

A: The resulting expression after distributing the $x$ term is $x(x^2 + 4x + 7) - 18$.

Q: How do we evaluate the final expression?

A: We can evaluate the final expression by substituting the value of $x$.

Q: What is the value of the polynomial $3x^3 - x^2 + 7x + 100$?

A: The value of the polynomial $3x^3 - x^2 + 7x + 100$ is $90$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $90$ feet.

Q: What is the main concept used to solve the problem?

A: The main concept used to solve the problem is algebraic manipulation, specifically factoring and distributing polynomials.

Q: What is the importance of factoring and distributing polynomials in solving the problem?

A: Factoring and distributing polynomials are essential in simplifying the polynomial expression and finding the value of the polynomial.

Q: What is the final result of the problem?

A: The final result of the problem is the distance between the two buildings, which is $90$ feet.

Q: What is the significance of the problem?

A: The problem is significant because it requires the application of algebraic manipulation to solve a real-world problem.

Q: What is the relevance of the problem to real-world scenarios?

A: The problem is relevant to real-world scenarios because it involves finding the distance between two buildings, which is a common problem in architecture and engineering.

Q: What is the key takeaway from the problem?

A: The key takeaway from the problem is the importance of algebraic manipulation in solving polynomial expressions and finding the value of polynomials.

Q: What is the final conclusion of the problem?

A: The final conclusion of the problem is that the distance between the two buildings is $90$ feet.