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Worksheets[ME] Division of Polynomials, Remainder and Factor Theorem
Total questions: 15
Worksheet time: 26mins
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.
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.
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.
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.
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.
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).
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.
What are the quotient and remainder when the polynomial P(x)=2x3−5x2+x−7 is divided by (x−3)?
Quotient: 2x2+x−2 , Remainder: -13
Quotient: 2x2−5x+1 , Remainder: -7
Quotient: 2x2+x+4 , Remainder: 5
Quotient: 2x2−11x+34 , Remainder: -109
What is the quotient when the polynomial P(x)= 4x3−6x2+2x−1 is divided by (2x−1)?
2x2+2x−1
2x2−2x−1
2x2−2x+1
2x2−2x
What is the remainder when the polynomial P(x)=3x3−2x2+5x−4 is divided by (x−1)?
Is (x−2) a factor of the polynomial P(x)=x3−4x2+5x−2 ?
Yes
No
A polynomial P(x)=2x4+3x3−4x+5 is given. Apply the Factor Theorem to determine if (2x−1) is a factor of P(x). If it isn't, explain the reason and state the remainder. (Upload the clear photo of your solution in the submission tab below).
If the polynomial P(x)=2x3−kx2+5x−1 is divided by x−2, the remainder is 11. What's the value of k?
The polynomial P(x)=x3+2x2+ax+b has a factor of (x−1). If the remainder when P(x) is divided by (x+2) is −12, what is the value of P(x) when x=2?
A civil engineer is designing a new roller coaster. The profile of a section of the track is modeled by the polynomial function H(x)=x4−10x3+35x2−50x+24 , where H(x) is the height of the track in meters and x is the horizontal distance in hundreds of meters from the starting point. The engineer knows that the track must have a constant height of zero meters at a certain point for safety reasons, which means the track must be on the ground. The design team has determined that the track will touch the ground at x=1, x=2, and x=3 hundred meters from the start. Use synthetic division to verify that the track is indeed at ground level at x=1 and then determine if there are any other possible ground-level points.
