[ME] Division of Polynomials, Remainder and Factor Theorem

Which of the following is a key limitation of synthetic division compared to long polynomial division?

• It cannot be used for polynomials with a degree greater than 3.

• It can only be used when the divisor is a linear polynomial of the form (x−c).

• It can only be used to divide polynomials with real coefficients.

• It does not provide the remainder of the division.

Answer explanation

This is the primary limitation. Synthetic division is a shortcut method that only works when the divisor is a linear binomial of the form (x−c), unlike long division which can handle divisors of any degree.

When you perform synthetic division on a polynomial P(x) using a value of c, and the remainder is zero, what is the most significant conclusion you can draw?

• The polynomial must have a degree of 2.

• The divisor is a factor of the polynomial.

• The polynomial is now a monomial.

• The quotient will be a constant term.

Answer explanation

A remainder of zero is the definition of a factor. This is the main principle of the Factor Theorem, which is directly applicable to the result of synthetic division.

Consider the synthetic division setup below for dividing a polynomial P(x) by (x−4). What does the last number in the bottom row represent?

• The value of the polynomial at x=−4.

• The constant term of the quotient.

• The constant term of the original polynomial.

• The remainder of the division.

Answer explanation

In a synthetic division setup, the final number in the bottom row after all calculations are completed is always the remainder of the division.

The Remainder Theorem states that when a polynomial P(x) is divided by a linear factor (x−c), the remainder is P(c). What is the most significant conceptual advantage of this theorem?

• It proves that polynomial division is always possible.

• It links the value of a function at a specific point to a polynomial's remainder.

• It allows us to find the roots of any polynomial without calculation.

Answer explanation

This is the core conceptual strength of the theorem. It establishes a direct and simple relationship between evaluating a polynomial at a number and finding the remainder when it's divided by a specific linear factor.

Given that the remainder is 5 when a polynomial P(x) is divided by (x−2), what can be concluded based on the Remainder Theorem?

• The value of the polynomial at x=2 is 5.

• The value of the polynomial at x=5 is 2.

• (x−2) is a factor of P(x).

• The value of the polynomial at x=−2 is 5.

Answer explanation

This is a direct application of the Remainder Theorem. The remainder when P(x) is divided by (x−2) is equal to P(2), so if the remainder is 5, then P(2) must be 5.

The Factor Theorem is a specific application of the Remainder Theorem. Which of the following statements best describes the relationship between the two theorems?

• The Remainder Theorem can only be used when the divisor is a linear factor, but the Factor Theorem can be used for any polynomial divisor.

• The Factor Theorem is used to find the remainder of a polynomial, while the Remainder Theorem is used to find a factor.

• The Factor Theorem applies to polynomials of any degree, whereas the Remainder Theorem only applies to cubic polynomials.

• The Factor Theorem states that a polynomial's remainder is zero, while the Remainder Theorem states the remainder is P(a).

Answer explanation

This statement is true. The Factor Theorem is a special case of the Remainder Theorem where the remainder P(a) is exactly zero, indicating that (x−a) is a factor.

Given a polynomial P(x), if we know that P(a)=0, which of the following is an immediate and direct conclusion based on the Factor Theorem?

• (x+a) is a factor of the polynomial P(x).

• The remainder when P(x) is divided by (x−a) is 0.

• (x−a) is a factor of the polynomial P(x).

• The value x=a is the only root of the polynomial.

Answer explanation

This is the direct conclusion from the Factor Theorem. If a value 'a' makes a polynomial equal to zero, then (x−a) must be one of its factors.

Answer explanation

This result comes from the correct synthetic division process. The coefficients of the quotient are 2, 1, and 4, and the final value is the remainder, 5. The quotient is of degree 2 since the original polynomial had a degree of 3.

Answer explanation

First, you must find the root of the divisor by setting 2x−1=0, which gives x=1/2. Performing synthetic division with the polynomial's coefficients (4, -6, 2, -1) and this root yields a bottom row of 4, -4, 0, -1. The first three numbers, 4, -4, and 0, are the coefficients of the quotient. Since the divisor's leading coefficient is 2, you must divide these coefficients by 2 to get the correct quotient: 4/2=2, −4/2=−2, and 0/2=0. This gives a quotient of 2x2−2x

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