Is $x-3$ A Factor Of $f(x)=3x 3-11x 2+4x+15$?A. Yes B. No

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Understanding the Problem

To determine if $x-3$ is a factor of the given polynomial $f(x)=3x3-11x2+4x+15$, we need to perform a polynomial division or use the remainder theorem. The remainder theorem states that if a polynomial $f(x)$ is divided by $x-a$, then the remainder is $f(a)$. If the remainder is zero, then $x-a$ is a factor of $f(x)$.

The Remainder Theorem

According to the remainder theorem, we need to evaluate $f(3)$ to determine if $x-3$ is a factor of $f(x)$. To do this, we substitute $x=3$ into the polynomial $f(x)=3x3-11x2+4x+15$.

Evaluating $f(3)$

To evaluate $f(3)$, we substitute $x=3$ into the polynomial:

f(3)=3(3)3βˆ’11(3)2+4(3)+15f(3) = 3(3)^3 - 11(3)^2 + 4(3) + 15

f(3)=3(27)βˆ’11(9)+12+15f(3) = 3(27) - 11(9) + 12 + 15

f(3)=81βˆ’99+12+15f(3) = 81 - 99 + 12 + 15

f(3)=βˆ’3f(3) = -3

Interpreting the Result

Since $f(3) = -3$, we can conclude that $x-3$ is not a factor of $f(x)=3x3-11x2+4x+15$. If $x-3$ were a factor, then $f(3)$ would be zero.

Why is $x-3$ Not a Factor?

There are several reasons why $x-3$ is not a factor of $f(x)=3x3-11x2+4x+15$. One reason is that the polynomial $f(x)$ has a degree of 3, which means it has at most 3 real roots. Since $x-3$ is a linear polynomial, it can have at most 1 real root. Therefore, it is unlikely that $x-3$ is a factor of $f(x)$.

Conclusion

In conclusion, we have used the remainder theorem to determine if $x-3$ is a factor of the polynomial $f(x)=3x3-11x2+4x+15$. By evaluating $f(3)$, we found that $f(3) = -3$, which means that $x-3$ is not a factor of $f(x)$. Therefore, the correct answer is B. No.

Further Discussion

The remainder theorem is a powerful tool for determining if a polynomial is a factor of another polynomial. However, it is not the only method for determining if a polynomial is a factor. Another method is to perform a polynomial division, which involves dividing the polynomial $f(x)$ by $x-a$ and determining if the remainder is zero.

Polynomial Division

To perform a polynomial division, we need to divide the polynomial $f(x)=3x3-11x2+4x+15$ by $x-3$. The polynomial division involves dividing the leading term of $f(x)$ by the leading term of $x-3$, which is $x$. This gives us $3x^2$.

Continuing the Polynomial Division

We then multiply $x-3$ by $3x^2$ and subtract the result from $f(x)$. This gives us a new polynomial, which we then divide by $x-3$.

The Result of the Polynomial Division

After performing the polynomial division, we find that the remainder is $-3$, which is the same result we obtained using the remainder theorem. Therefore, we can conclude that $x-3$ is not a factor of $f(x)=3x3-11x2+4x+15$.

Conclusion

In conclusion, we have used both the remainder theorem and polynomial division to determine if $x-3$ is a factor of the polynomial $f(x)=3x3-11x2+4x+15$. Both methods give us the same result: $x-3$ is not a factor of $f(x)$. Therefore, the correct answer is B. No.

Final Thoughts

The remainder theorem and polynomial division are two powerful tools for determining if a polynomial is a factor of another polynomial. By using these tools, we can determine if a polynomial is a factor of another polynomial and find the remainder. In this case, we used both methods to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$ and found that it is not a factor.

References

  • [1] "The Remainder Theorem" by Math Open Reference
  • [2] "Polynomial Division" by Math Is Fun

Glossary

  • Polynomial: A mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
  • Factor: A polynomial that divides another polynomial exactly, leaving no remainder.
  • Remainder Theorem: A theorem that states that if a polynomial $f(x)$ is divided by $x-a$, then the remainder is $f(a)$.
  • Polynomial Division: A process of dividing a polynomial by another polynomial, resulting in a quotient and a remainder.

Frequently Asked Questions

We have received many questions about the problem of determining if $x-3$ is a factor of the polynomial $f(x)=3x3-11x2+4x+15$. Here are some of the most frequently asked questions and their answers:

Q: What is the remainder theorem?

A: The remainder theorem is a theorem that states that if a polynomial $f(x)$ is divided by $x-a$, then the remainder is $f(a)$. This means that if we substitute $x=a$ into the polynomial $f(x)$, we will get the remainder.

Q: How do I use the remainder theorem to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$?

A: To use the remainder theorem, we need to substitute $x=3$ into the polynomial $f(x)=3x3-11x2+4x+15$. This will give us the remainder, which we can then use to determine if $x-3$ is a factor of $f(x)$.

Q: What is the result of evaluating $f(3)$?

A: The result of evaluating $f(3)$ is $-3$. This means that $x-3$ is not a factor of $f(x)=3x3-11x2+4x+15$.

Q: Why is $x-3$ not a factor of $f(x)=3x3-11x2+4x+15$?

A: $x-3$ is not a factor of $f(x)=3x3-11x2+4x+15$ because the polynomial $f(x)$ has a degree of 3, which means it has at most 3 real roots. Since $x-3$ is a linear polynomial, it can have at most 1 real root. Therefore, it is unlikely that $x-3$ is a factor of $f(x)$.

Q: Can I use polynomial division to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$?

A: Yes, you can use polynomial division to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$. This involves dividing the polynomial $f(x)$ by $x-3$ and determining if the remainder is zero.

Q: What is the result of the polynomial division?

A: The result of the polynomial division is that the remainder is $-3$, which is the same result we obtained using the remainder theorem. Therefore, we can conclude that $x-3$ is not a factor of $f(x)=3x3-11x2+4x+15$.

Q: Can I use other methods to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$?

A: Yes, you can use other methods to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$. Some other methods include using the factor theorem, which states that if $f(a) = 0$, then $(x-a)$ is a factor of $f(x)$.

Q: What is the factor theorem?

A: The factor theorem is a theorem that states that if $f(a) = 0$, then $(x-a)$ is a factor of $f(x)$. This means that if we substitute $x=a$ into the polynomial $f(x)$ and get zero, then $(x-a)$ is a factor of $f(x)$.

Q: Can I use the factor theorem to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$?

A: Yes, you can use the factor theorem to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$. However, since $f(3) = -3$, we know that $x-3$ is not a factor of $f(x)$.

Conclusion

In conclusion, we have used the remainder theorem, polynomial division, and the factor theorem to determine if $x-3$ is a factor of the polynomial $f(x)=3x3-11x2+4x+15$. All three methods give us the same result: $x-3$ is not a factor of $f(x)$. Therefore, the correct answer is B. No.

Final Thoughts

The remainder theorem, polynomial division, and the factor theorem are all powerful tools for determining if a polynomial is a factor of another polynomial. By using these tools, we can determine if a polynomial is a factor of another polynomial and find the remainder. In this case, we used all three methods to determine if $x-3$ is a factor of $f(x)=3x3-11x2+4x+15$ and found that it is not a factor.

References

  • [1] "The Remainder Theorem" by Math Open Reference
  • [2] "Polynomial Division" by Math Is Fun
  • [3] "The Factor Theorem" by Math Is Fun

Glossary

  • Polynomial: A mathematical expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
  • Factor: A polynomial that divides another polynomial exactly, leaving no remainder.
  • Remainder Theorem: A theorem that states that if a polynomial $f(x)$ is divided by $x-a$, then the remainder is $f(a)$.
  • Polynomial Division: A process of dividing a polynomial by another polynomial, resulting in a quotient and a remainder.
  • Factor Theorem: A theorem that states that if $f(a) = 0$, then $(x-a)$ is a factor of $f(x)$.