Which of the following statements is not true regarding the division of polynomials?   Question 18 options:

Which of the following statements is not true regarding the division of polynomials? Question 18 options: a) Terms should be arranged in descending order of degree in both the divisor and the dividend. b) If the remainder of polynomial division is zero, both the divisor and the quotient are factors of the dividend. c) A polynomial can only be divided by a polynomial of the same degree or less. d) Polynomial division is very different than the way numbers are divided.

The statement that is not true regarding the division of polynomials is option c) “A polynomial can only be divided by a polynomial of the same degree or less.”

Polynomial division allows for the division of a polynomial by a polynomial of a different degree. This is one of the key differences between polynomial division and regular number division. In number division, the divisor must be of the same magnitude or smaller than the dividend. However, in polynomial division, there is no such constraint on the degrees of the polynomials involved.

To understand why option c) is not true, let’s look at an example. Consider the polynomial division of (x^3 + 2x^2 – 3x + 1) by (x^2 – x + 2). Here, the degree of the divisor (x^2 – x + 2) is 2, while the degree of the dividend (x^3 + 2x^2 – 3x + 1) is 3. Thus, the statement that a polynomial can only be divided by a polynomial of the same degree or less is not true.

In fact, polynomial division allows us to divide a polynomial by a polynomial of any degree, as long as the divisor is nonzero. The result of this division may be a quotient polynomial and a remainder polynomial. The quotient polynomial represents the whole number of times the divisor can be divided into the dividend, while the remainder polynomial represents any leftover terms that cannot be divided evenly.

Polynomial division is a fundamental concept in algebra and is used in many areas of mathematics and other sciences. It is particularly important in polynomial factorization and finding roots of polynomials. By dividing a polynomial by its factors, we can determine its root values, which are the values that make the polynomial equal to zero.

In conclusion, option c) “A polynomial can only be divided by a polynomial of the same degree or less” is not true. Polynomial division allows for the division of a polynomial by a polynomial of any degree, as long as the divisor is nonzero. This is a key difference between polynomial division and regular number division.