For What Values Of $a$ And $b$ Are The Following Polynomials Equal?${ \begin{array}{l} f(x) = 2x^3 + 4x^2 - (2a + B)x + 3 \ g(x) = 2x^3 + (a + B)x^2 - 7x + 3 \end{array} }$

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Introduction

In mathematics, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When comparing two polynomials, we need to determine if they are equal for all values of the variable. In this article, we will explore the values of $a$ and $b$ for which the given polynomials $f(x)$ and $g(x)$ are equal.

The Polynomials

We are given two polynomials:

$$f(x) = 2x^3 + 4x^2 - (2a + b)x + 3$$

$$g(x) = 2x^3 + (a + b)x^2 - 7x + 3$$

To determine if these polynomials are equal, we need to compare their coefficients and constant terms.

Comparing Coefficients

When comparing the coefficients of the two polynomials, we can see that the coefficients of the $x^3$ term are the same, which is 2. The coefficients of the $x^2$ term are also different, with $f(x)$ having a coefficient of 4 and $g(x)$ having a coefficient of $(a + b)$. The coefficients of the $x$ term are also different, with $f(x)$ having a coefficient of $-(2a + b)$ and $g(x)$ having a coefficient of -7.

Equating Coefficients

To determine the values of $a$ and $b$ for which the polynomials are equal, we need to equate the coefficients of the corresponding terms. We can start by equating the coefficients of the $x^2$ term:

$$4 = a + b$$

Equating the Constant Terms

Next, we can equate the constant terms of the two polynomials:

$$3 = 3$$

This equation is always true, so we don't need to do anything with it.

Equating the Coefficients of the $x$ Term

Now, we can equate the coefficients of the $x$ term:

$$-(2a + b) = -7$$

Solving the System of Equations

We now have a system of two equations with two unknowns:

$$4 = a + b$$

$$-(2a + b) = -7$$

We can solve this system of equations using substitution or elimination. Let's use substitution.

Solving for $a$

We can solve the first equation for $a$:

$$a = 4 - b$$

Substituting into the Second Equation

Now, we can substitute this expression for $a$ into the second equation:

$$-(2(4 - b) + b) = -7$$

Simplifying the Equation

We can simplify this equation:

$$-8 + 2b - b = -7$$

Combining Like Terms

We can combine like terms:

$$-8 + b = -7$$

Solving for $b$

We can solve for $b$:

$$b = -8 + 7$$

$$b = -1$$

Finding the Value of $a$

Now that we have the value of $b$, we can find the value of $a$:

$$a = 4 - b$$

$$a = 4 - (-1)$$

$$a = 5$$

Conclusion

In conclusion, the values of $a$ and $b$ for which the polynomials $f(x)$ and $g(x)$ are equal are $a = 5$ and $b = -1$.

Final Answer

The final answer is $\boxed{a = 5, b = -1}$.

Discussion

This problem is a great example of how to compare polynomials and find the values of the coefficients for which they are equal. It requires a good understanding of algebra and the ability to solve systems of equations.

Related Problems

If you want to practice more problems like this, you can try the following:

• Compare the polynomials $f(x) = 2x^3 + 3x^2 - 4x + 1$ and $g(x) = 2x^3 + (a + b)x^2 - 7x + 3$.
• Find the values of $a$ and $b$ for which the polynomials $f(x) = x^2 + (a + b)x + 1$ and $g(x) = x^2 + (a - b)x + 1$ are equal.

References

• [1] "Algebra" by Michael Artin
• [2] "Polynomials" by Wolfram MathWorld

Keywords

• Polynomials
• Algebra
• Equations
• Coefficients
• Constant terms
• Systems of equations
• Substitution
• Elimination

Category

• Mathematics
• Algebra
• Polynomials
  

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Introduction

In our previous article, we explored the values of $a$ and $b$ for which the given polynomials $f(x)$ and $g(x)$ are equal. In this article, we will answer some frequently asked questions related to this topic.

Q: What are the main differences between the two polynomials?

A: The main differences between the two polynomials are the coefficients of the $x^2$ term and the $x$ term. The coefficient of the $x^2$ term in $f(x)$ is 4, while the coefficient of the $x^2$ term in $g(x)$ is $(a + b)$. The coefficient of the $x$ term in $f(x)$ is $-(2a + b)$, while the coefficient of the $x$ term in $g(x)$ is -7.

Q: How do we equate the coefficients of the corresponding terms?

A: To equate the coefficients of the corresponding terms, we need to set up a system of equations. We can start by equating the coefficients of the $x^2$ term:

$$4 = a + b$$

Next, we can equate the coefficients of the $x$ term:

$$-(2a + b) = -7$$

Q: How do we solve the system of equations?

A: We can solve the system of equations using substitution or elimination. Let's use substitution. We can solve the first equation for $a$:

$$a = 4 - b$$

Then, we can substitute this expression for $a$ into the second equation:

$$-(2(4 - b) + b) = -7$$

Q: What is the final answer?

A: The final answer is $\boxed{a = 5, b = -1}$.

Q: Can you provide more examples of polynomials that can be compared?

A: Yes, here are a few examples:

• Compare the polynomials $f(x) = 2x^3 + 3x^2 - 4x + 1$ and $g(x) = 2x^3 + (a + b)x^2 - 7x + 3$.
• Find the values of $a$ and $b$ for which the polynomials $f(x) = x^2 + (a + b)x + 1$ and $g(x) = x^2 + (a - b)x + 1$ are equal.

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

A: Some common mistakes to avoid when comparing polynomials include:

• Not equating the coefficients of the corresponding terms.
• Not solving the system of equations correctly.
• Not checking the constant terms.

Q: Can you provide some references for further reading?

A: Yes, here are a few references for further reading:

• [1] "Algebra" by Michael Artin
• [2] "Polynomials" by Wolfram MathWorld

Q: What are some keywords related to this topic?

A: Some keywords related to this topic include:

• Polynomials
• Algebra
• Equations
• Coefficients
• Constant terms
• Systems of equations
• Substitution
• Elimination

Q: What is the category of this topic?

A: The category of this topic is Mathematics, specifically Algebra and Polynomials.