Dividing polynomials

Multiple Choice

Which of the following methods can be used to divide polynomial expressions by linear factors?

• Long division

• Synthetic division

• Factoring by grouping

• Completing the square

Open Ended

Why is it important to learn how to divide polynomials using long division and synthetic division?

Multiple Choice

Which part of the polynomial division setup represents the divisor?

• x + 2

• 3x + 5

• 3x^2 + 11x + 10

• x^2 + 2x

Open Ended

Explain the steps involved in dividing polynomials using long division.

Open Ended

Describe how the process of dividing polynomials is similar to dividing numbers using long division.

Multiple Choice

Which of the following expressions correctly represents the relationship between dividend, divisor, quotient, and remainder in polynomial division?

• dividend = divisor × quotient + remainder

• dividend = quotient × divisor + remainder

• divisor = dividend × quotient + remainder

• dividend = remainder × divisor + quotient

Fill in the Blanks

When dividing polynomials, if a term is missing in the dividend, what coefficient should be used for that term?

Multiple Select

Select all statements that are true about the polynomial division shown in Example 1 on pages 154 and 155.

• The remainder is 0 in the second example.

• The divisor is a factor of the dividend in the second example.

• The dividend must always have all degree terms present.

• The quotient is always a polynomial.

Open Ended

Use long division to divide the polynomials. Then write the dividend in terms of the quotient and remainder. a. x^3 - 6x^2 + 11x - 6 divided by x^2 - 4x + 3 b. 16x^4 - 85 divided by 4x^2 + 9

Multiple Choice

Which of the following is true about synthetic division?

• It can be used to divide any polynomial by any divisor.

• It only works when dividing by a linear polynomial of the form x - a.

• It is used to multiply polynomials.

• It only works for quadratic divisors.

Fill in the Blanks

What is the remainder when 2x^3 - 7x^2 - 4 is divided by x - 3 using synthetic division?

Open Ended

Use synthetic division to divide 3x^3 - 5x + 10 by x - 1.

Open Ended

Explain why synthetic division cannot be used to divide by a quadratic polynomial.

Multiple Select

Select all statements that are correct about synthetic division.

• Synthetic division only works when dividing by a linear polynomial.

• The degree of the divisor must be 1.

• Synthetic division can be used for quadratic divisors.

• Synthetic division simplifies the division process.

Multiple Choice

Which method is generally faster for dividing polynomials: long division or synthetic division?

• Long division

• Synthetic division

• Both are equally fast

• It depends on the degree of the polynomial

Fill in the Blanks

Divide (4x^2 + 4x + 5) by (x + 5) using long division. What is the quotient?

Open Ended

Compare the answers obtained from Practice 1 and Practice 2. Are the quotients the same when using long division and synthetic division for (4x^2 + 4x + 5) ÷ (x + 5)? Why or why not?

Fill in the Blanks

Divide (4x^2 + 4x + 5) by (x + 5) using synthetic division. What is the quotient?

Multiple Choice

How is the value of P(a) related to the remainder when P(x) is divided by (x - a)?

• P(a) is always zero

• P(a) is the remainder

• P(a) is the quotient

• P(a) is the divisor

Multiple Select

Which of the following statements are true about the Remainder Theorem and Factor Theorem?

• The remainder theorem states that the remainder is P(a) when dividing by x-a.

• The factor theorem states that x-a is a factor if and only if P(a)=0.

• The remainder theorem applies only to linear divisors.

• The factor theorem applies only to quadratic polynomials.

Fill in the Blanks

Polynomial expressions can be divided by linear factors using

division or synthetic division.

Open Ended

How can synthetic division be used to rewrite rational expressions?