What Is The Remainder When $3x^2 - 7x + 5$ Is Divided By $x + 5$?Enter Your Answer As An Integer, Like This: 42.

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Introduction

When a polynomial is divided by another polynomial, the remainder is a polynomial of degree less than the divisor. In this case, we are asked to find the remainder when the polynomial $3x^2 - 7x + 5$ is divided by the polynomial $x + 5$. To find the remainder, we can use the Remainder Theorem, which states that if a polynomial $f(x)$ is divided by $x - a$, then the remainder is equal to $f(a)$. In this case, we can rewrite the divisor as $x - (-5)$, so we can use the Remainder Theorem to find the remainder.

The Remainder Theorem

The Remainder Theorem is a powerful tool for finding the remainder of a polynomial division. It states that if a polynomial $f(x)$ is divided by $x - a$, then the remainder is equal to $f(a)$. This means that we can find the remainder by simply plugging in the value of $a$ into the polynomial. In this case, we want to find the remainder when $3x^2 - 7x + 5$ is divided by $x + 5$, so we can rewrite the divisor as $x - (-5)$ and use the Remainder Theorem.

Applying the Remainder Theorem

To find the remainder, we need to plug in the value of $a$ into the polynomial. In this case, $a = -5$, so we need to plug in $-5$ into the polynomial $3x^2 - 7x + 5$. This gives us:

$$3(-5)^2 - 7(-5) + 5$$

Evaluating the Expression

Now we need to evaluate the expression $3(-5)^2 - 7(-5) + 5$. To do this, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent: $(-5)^2 = 25$
2. Multiply: $3(25) = 75$
3. Multiply: $-7(-5) = 35$
4. Add: $75 + 35 = 110$
5. Add: $110 + 5 = 115$

Conclusion

Therefore, the remainder when $3x^2 - 7x + 5$ is divided by $x + 5$ is $\boxed{115}$.

Example

Let's try another example to see how the Remainder Theorem works. Suppose we want to find the remainder when the polynomial $x^2 + 4x + 4$ is divided by $x + 2$. We can rewrite the divisor as $x - (-2)$ and use the Remainder Theorem.

Applying the Remainder Theorem

To find the remainder, we need to plug in the value of $a$ into the polynomial. In this case, $a = -2$, so we need to plug in $-2$ into the polynomial $x^2 + 4x + 4$. This gives us:

$$(-2)^2 + 4(-2) + 4$$

Evaluating the Expression

Now we need to evaluate the expression $(-2)^2 + 4(-2) + 4$. To do this, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent: $(-2)^2 = 4$
2. Multiply: $4(-2) = -8$
3. Add: $4 + (-8) = -4$
4. Add: $-4 + 4 = 0$

Conclusion

Therefore, the remainder when $x^2 + 4x + 4$ is divided by $x + 2$ is $\boxed{0}$.

The Remainder Theorem in Real-World Applications

The Remainder Theorem has many real-world applications. For example, it can be used to find the remainder of a polynomial division in a variety of fields, such as engineering, physics, and computer science. It can also be used to find the remainder of a polynomial division in a variety of contexts, such as finding the remainder of a polynomial division in a mathematical competition or finding the remainder of a polynomial division in a real-world problem.

Conclusion

In conclusion, the Remainder Theorem is a powerful tool for finding the remainder of a polynomial division. It can be used to find the remainder of a polynomial division in a variety of fields and contexts. By using the Remainder Theorem, we can find the remainder of a polynomial division in a variety of situations, from mathematical competitions to real-world problems.

References

• [1] "The Remainder Theorem" by Math Open Reference
• [2] "Polynomial Division" by Khan Academy
• [3] "The Remainder Theorem" by Wolfram MathWorld

Further Reading

• [1] "Polynomial Division" by MIT OpenCourseWare
• [2] "The Remainder Theorem" by University of California, Berkeley
• [3] "Polynomial Division" by University of Michigan

FAQs

• Q: What is the Remainder Theorem? A: The Remainder Theorem is a powerful tool for finding the remainder of a polynomial division.
• Q: How do I use the Remainder Theorem? A: To use the Remainder Theorem, you need to plug in the value of $a$ into the polynomial.
• Q: What are some real-world applications of the Remainder Theorem? A: The Remainder Theorem has many real-world applications, including finding the remainder of a polynomial division in a variety of fields and contexts.
  

Q: What is the Remainder Theorem?

A: The Remainder Theorem is a powerful tool for finding the remainder of a polynomial division. It states that if a polynomial $f(x)$ is divided by $x - a$, then the remainder is equal to $f(a)$.

Q: How do I use the Remainder Theorem?

A: To use the Remainder Theorem, you need to plug in the value of $a$ into the polynomial. This means that you need to substitute the value of $a$ into the polynomial and simplify the expression.

Q: What are some common mistakes to avoid when using the Remainder Theorem?

A: Some common mistakes to avoid when using the Remainder Theorem include:

• Not substituting the value of $a$ into the polynomial correctly
• Not simplifying the expression correctly
• Not checking the degree of the polynomial and the divisor

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a non-linear divisor?

A: No, the Remainder Theorem only works for polynomial divisions with a linear divisor. If you have a polynomial division with a non-linear divisor, you will need to use a different method to find the remainder.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a complex divisor?

A: Yes, the Remainder Theorem can be used to find the remainder of a polynomial division with a complex divisor. However, you will need to use complex numbers and follow the same steps as you would with real numbers.

Q: How do I know if the remainder is a polynomial or a constant?

A: If the degree of the remainder is less than the degree of the divisor, then the remainder is a polynomial. If the degree of the remainder is equal to the degree of the divisor, then the remainder is a constant.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a rational divisor?

A: Yes, the Remainder Theorem can be used to find the remainder of a polynomial division with a rational divisor. However, you will need to use rational numbers and follow the same steps as you would with real numbers.

Q: How do I check if the remainder is correct?

A: To check if the remainder is correct, you can use the following steps:

• Plug in the value of $a$ into the polynomial and simplify the expression
• Check if the degree of the remainder is less than the degree of the divisor
• Check if the remainder is a polynomial or a constant

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor?

A: Yes, the Remainder Theorem can be used to find the remainder of a polynomial division with a polynomial divisor. However, you will need to use polynomial long division and follow the same steps as you would with real numbers.

Q: How do I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor?

A: To use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor, you will need to follow the same steps as you would with a linear divisor. However, you will need to use polynomial long division and simplify the expression.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a non-linear remainder?

A: No, the Remainder Theorem only works for polynomial divisions with a linear divisor. If you have a polynomial division with a polynomial divisor and a non-linear remainder, you will need to use a different method to find the remainder.

Q: How do I know if the remainder is a polynomial or a constant when using the Remainder Theorem with a polynomial divisor?

A: If the degree of the remainder is less than the degree of the divisor, then the remainder is a polynomial. If the degree of the remainder is equal to the degree of the divisor, then the remainder is a constant.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a complex remainder?

A: Yes, the Remainder Theorem can be used to find the remainder of a polynomial division with a polynomial divisor and a complex remainder. However, you will need to use complex numbers and follow the same steps as you would with real numbers.

Q: How do I check if the remainder is correct when using the Remainder Theorem with a polynomial divisor?

A: To check if the remainder is correct, you can use the following steps:

• Plug in the value of $a$ into the polynomial and simplify the expression
• Check if the degree of the remainder is less than the degree of the divisor
• Check if the remainder is a polynomial or a constant

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a rational remainder?

A: Yes, the Remainder Theorem can be used to find the remainder of a polynomial division with a polynomial divisor and a rational remainder. However, you will need to use rational numbers and follow the same steps as you would with real numbers.

Q: How do I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a rational remainder?

A: To use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a rational remainder, you will need to follow the same steps as you would with a linear divisor. However, you will need to use polynomial long division and simplify the expression.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a non-rational remainder?

A: No, the Remainder Theorem only works for polynomial divisions with a linear divisor. If you have a polynomial division with a polynomial divisor and a non-rational remainder, you will need to use a different method to find the remainder.

Q: How do I know if the remainder is a polynomial or a constant when using the Remainder Theorem with a polynomial divisor and a non-rational remainder?

A: If the degree of the remainder is less than the degree of the divisor, then the remainder is a polynomial. If the degree of the remainder is equal to the degree of the divisor, then the remainder is a constant.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a complex remainder and a non-rational remainder?

A: No, the Remainder Theorem only works for polynomial divisions with a linear divisor. If you have a polynomial division with a polynomial divisor and a complex remainder and a non-rational remainder, you will need to use a different method to find the remainder.

Q: How do I check if the remainder is correct when using the Remainder Theorem with a polynomial divisor and a complex remainder and a non-rational remainder?

A: To check if the remainder is correct, you can use the following steps:

• Plug in the value of $a$ into the polynomial and simplify the expression
• Check if the degree of the remainder is less than the degree of the divisor
• Check if the remainder is a polynomial or a constant

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a rational remainder and a non-rational remainder?

A: No, the Remainder Theorem only works for polynomial divisions with a linear divisor. If you have a polynomial division with a polynomial divisor and a rational remainder and a non-rational remainder, you will need to use a different method to find the remainder.

Q: How do I know if the remainder is a polynomial or a constant when using the Remainder Theorem with a polynomial divisor and a rational remainder and a non-rational remainder?

A: If the degree of the remainder is less than the degree of the divisor, then the remainder is a polynomial. If the degree of the remainder is equal to the degree of the divisor, then the remainder is a constant.

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a complex remainder and a rational remainder?

A: Yes, the Remainder Theorem can be used to find the remainder of a polynomial division with a polynomial divisor and a complex remainder and a rational remainder. However, you will need to use complex numbers and rational numbers and follow the same steps as you would with real numbers.

Q: How do I check if the remainder is correct when using the Remainder Theorem with a polynomial divisor and a complex remainder and a rational remainder?

A: To check if the remainder is correct, you can use the following steps:

• Plug in the value of $a$ into the polynomial and simplify the expression
• Check if the degree of the remainder is less than the degree of the divisor
• Check if the remainder is a polynomial or a constant

Q: Can I use the Remainder Theorem to find the remainder of a polynomial division with a polynomial divisor and a non-rational remainder and a rational remainder?

A: No, the Remainder Theorem only works for polynomial divisions with a linear divisor. If you have a polynomial division with a polynomial divisor and a non-rational remainder and a rational remainder, you will need to use a different method to find the remainder.

Q: How do I know if the remainder is a polynomial or a constant when using the Remainder Theorem with a polynomial divisor and a non-rational remainder and a rational remainder?

A: If the degree of the remainder is less than the degree of the divisor, then the remainder is a polynomial. If the degree of the remainder is equal to the degree of the divisor, then the remainder is a constant