Understanding Remainders in Polynomial Division

Oct 13, 2025 by windailys.com 103 views

Introduction

When dealing with polynomial expressions, understanding the properties of remainders is crucial in solving various mathematical problems. In this context, we are given a polynomial expression and asked to determine which statement is true if $m_i$ gets a remainder of 14 when dividing the given expressions. This problem requires a deep understanding of polynomial division, remainders, and the properties of polynomial expressions.

Understanding Remainders in Polynomial Division

In polynomial division, the remainder is the amount left over after dividing one polynomial by another. When a polynomial $f(x)$ is divided by another polynomial $g(x)$ , the remainder can be expressed as $f(x) = q(x) \cdot g(x) + r(x)$ , where $q(x)$ is the quotient and $r(x)$ is the remainder. The remainder $r(x)$ is a polynomial of degree less than the divisor $g(x)$ .

Analyzing the Given Expressions

We are given the polynomial expression $x^3 - x^2 + 3x - 4$ and asked to determine which statement is true if $m_i$ gets a remainder of 14 when dividing the given expressions. To approach this problem, we need to understand the properties of polynomial expressions and how remainders are affected when dividing by a polynomial.

Statement A: $f(3) = 14$

To determine if statement A is true, we need to evaluate the polynomial expression $f(x) = x^3 - x^2 + 3x - 4$ at $x = 3$ . Substituting $x = 3$ into the expression, we get:

$f(3) = (3)^3 - (3)^2 + 3(3) - 4$ $f(3) = 27 - 9 + 9 - 4$ $f(3) = 23$

Since $f(3) = 23$ and not 14, statement A is false.

Statement B: $f(3) = 14$

Statement B is identical to statement A, and we have already determined that statement A is false. Therefore, statement B is also false.

Statement C: $f(0.14) = 3$

To determine if statement C is true, we need to evaluate the polynomial expression $f(x) = x^3 - x^2 + 3x - 4$ at $x = 0.14$ . Substituting $x = 0.14$ into the expression, we get:

$f(0.14) = (0.14)^3 - (0.14)^2 + 3(0.14) - 4$ $f(0.14) = 0.0028 - 0.0196 + 0.42 - 4$ $f(0.14) = -3.6168$

Since $f(0.14) = -3.6168$ and not 3, statement C is false.

Statement D: $x - 315$ is a factor of $x^3 - x^2 + 3x - 4$

To determine if statement D is true, we need to check if $x - 315$ is a factor of the polynomial expression $x^3 - x^2 + 3x - 4$ . We can do this by using polynomial long division or synthetic division.

Using polynomial long division, we can divide the polynomial expression $x^3 - x^2 + 3x - 4$ by $x - 315$ . The result of the division is:

$x^2 + 316x + 1001$

Since the remainder is not zero, $x - 315$ is not a factor of the polynomial expression $x^3 - x^2 + 3x - 4$ . Therefore, statement D is false.

Conclusion

In conclusion, none of the statements A, B, C, or D are true. The polynomial expression $x^3 - x^2 + 3x - 4$ does not have a remainder of 14 when divided by any of the given expressions. The properties of polynomial expressions and remainders were used to analyze each statement and determine its validity.

Final Answer

The final answer is that none of the statements A, B, C, or D are true.

Introduction

In our previous article, we analyzed the properties of polynomial expressions and remainders to determine which statement is true if $m_i$ gets a remainder of 14 when dividing the given expressions. We found that none of the statements A, B, C, or D are true. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights.

Q&A

Q: What is the remainder theorem?

A: The remainder theorem states that if a polynomial $f(x)$ is divided by $x - a$ , then the remainder is equal to $f(a)$ .

Q: How do you determine if a polynomial is a factor of another polynomial?

A: To determine if a polynomial is a factor of another polynomial, you can use polynomial long division or synthetic division. If the remainder is zero, then the polynomial is a factor.

Q: What is the difference between a factor and a multiple?

A: A factor is a polynomial that divides another polynomial exactly, leaving no remainder. A multiple is a polynomial that is the product of another polynomial and a constant.

Q: Can a polynomial have multiple factors?

A: Yes, a polynomial can have multiple factors. For example, the polynomial $x^2 - 4$ has two factors: $(x - 2)$ and $(x + 2)$ .

Q: How do you determine the degree of a polynomial?

A: The degree of a polynomial is the highest power of the variable in the polynomial. For example, the polynomial $x^3 + 2x^2 - 3x + 1$ has a degree of 3.

Q: What is the difference between a polynomial and a rational expression?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression is an expression consisting of a polynomial divided by another polynomial.

Q: Can a polynomial have a remainder of 14 when divided by another polynomial?

A: Yes, a polynomial can have a remainder of 14 when divided by another polynomial. However, in the case of the given expressions, none of the statements A, B, C, or D are true.

Q: How do you evaluate a polynomial expression at a given value?

A: To evaluate a polynomial expression at a given value, you substitute the value into the expression and simplify.

Q: What is the significance of the remainder theorem in polynomial division?

A: The remainder theorem is significant in polynomial division because it allows us to determine the remainder of a polynomial when divided by another polynomial.

Conclusion

In conclusion, the Q&A section provides additional insights and clarifies the concepts related to polynomial expressions and remainders. We hope that this article has been helpful in understanding the properties of polynomial expressions and remainders.

Final Answer

The final answer is that none of the statements A, B, C, or D are true.