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Lectures
6 Videos

The Division Algorithm

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The Division Algorithm

Long Division Ex1

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3.39mins

Long Division Ex1

Long Division Ex2 Converting Rational Expression to Mixed Form

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Long Division Ex2 Converting Rational Expression to Mixed Form

Long Division Ex3 Dividing with Quotient of Degree 2

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Long Division Ex3 Dividing with Quotient of Degree 2

Introduction to Synthetic Division

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Introduction to Synthetic Division

Synthetic Division Example

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Synthetic Division Example

Solutions
51 Videos

Divide `x^4 -16x^3 + 4x^2 + 10x -11`

by `x - 2`

.

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1.32mins

Q1ai

Divide `x^4 -16x^3 + 4x^2 + 10x -11`

by `x +4`

.

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Q1aii

Divide `x^4 -16x^3 + 4x^2 + 10x -11`

by `x -1`

.

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1.06mins

Q1aiii

Are any of the binomials below factors of `x^4 -16x^3 +4x^2 +10x -11`

? Explain.

`x- 2`

`x+ 4`

`x-1`

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Q1b

State the degree of the quotient for each of the following division statements, if possible.

`(x^4 - 15x^3 + 2x^2 + 12x -10) \div (x^2 -4)`

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Q2a

State the degree of the quotient for each of the following division statements, if possible.

`(5x^3 -4x^2 + 3x -4) \div ( x +3)`

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Q2b

State the degree of the quotient for each of the following division statements, if possible.

`(x^4 - 7x^3 + 2x^2 + 9x)\div (x^3 -x^2 + 2x + 1)`

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Q2c

State the degree of the quotient for each of the following division statements, if possible.

`(2x^2 +5x -4)\div (x^4 + 3x^3 - 5x^2 +4x - 2)`

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Q2d

Complete the divisions

`(x^4 -15x^3 + 2x^3 + 12x -10) \div (x^2 -4)`

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Q3a

Complete the divisions

`(5x^3 -4x^2 + 3x -4) \div ( x +3)`

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Q3b

Complete the divisions

`(x^4 - 7x^3 + 2x^2 + 9x)\div (x^3 -x^2 + 2x + 1)`

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Q3c

Is this expression solvable? Explain.

`(2x^2 +5x -4)\div (x^4 + 3x^3 - 5x^2 +4x - 2)`

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Q3d

Complete the following table.

a) Given `P(x) = 2x^3 -5x^2 + 8x + 4, D(x) = x+ 3, Q(x) =2x^2 -11x + 41`

, find `R(x)`

.

b) Given `D(x) = 2x + 4, Q(x) =3x^2 -5x + 8, R(x) = -3`

, find `P(x)`

.

c) Given `P(x) = 6x^4 + 2x^3 + 3x^2 -11x - 9, Q(x) =2x^3 + x -4, R(x) = -5`

, find `D(x)`

.

a) Given `P(x) = 3x^3 + x^2 -6x + 16, D(x) = x +2, R(x) =8`

, find `Q(x)`

.

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Q4

Calculate each of the following using long division.

`(x^3 - 2x + 1)\div (x - 4)`

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Q5a

Calculate each of the following using long division.

`(x^3 + 2x^2 - 6x + 1) \div ( x +2)`

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Q5b

Calculate each of the following using long division.

`(2x^3 + 5x^2 - 4x -5)\div (2x + 1)`

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Q5c

Calculate each of the following using long division.

`(x^4 + 3x^3 - 2x^2 + 5x - 1)\div (x^2 + 7)`

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Q5d

Calculate each of the following using long division.

`(x^4 + 6x^3 - 8x +12)\div (x^3 -x^2 -x + 1)`

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Q5e

Calculate each of the following using synthetic division.

`(x^3 -7x - 6)\div(x - 3)`

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Q6a

Calculate each of the following using synthetic division.

`(2x^3 - 7x^2 -7x + 19)\div(x - 1)`

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Q6b

Calculate each of the following using synthetic division.

`(6x^4 + 13x^3 - 34x^2 - 47x + 28)\div(x + 3)`

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Q6c

Calculate each of the following using synthetic division.

`(2x^3 +x^2 -22x + 20)\div(2x -3)`

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Q6d

Calculate each of the following using synthetic division.

`(12x^4 - 56x^3 + 59x^2 + 9x -18)\div(2x + 1)`

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Q6e

Each divisor was divided into another polynomial, resulting in the given quotient and remainder. Find the other polynomial (the dividend).

divisor: `x+ 10`

, quotient: `x^2 - 6x + 9`

, remainder: `-1`

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Q7a

Each divisor was divided into another polynomial, resulting in the given quotient and remainder. Find the other polynomial (the dividend).

divisor: `5x + 2`

, quotient: `x^3 + 4x^2 -5x + 6`

, remainder: `x - 2`

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Q7c

Determine the remainder, `r`

, to make each multiplication statement true.

`(2x -3)(3x +5) + r = 6x^2 +x + 5`

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Q8a

Determine the remainder, `r`

, to make each multiplication statement true.

`(x+ 3)(x +5) + r = x^2 + 9x - 7`

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Q8b

Determine the remainder, `r`

, to make each multiplication statement true.

`(x+ 3)(x^2 - 1) + r =x^3 + 3x^2 -x - 3`

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Q8c

Determine the remainder, `r`

, to make each multiplication statement true.

`(x^2 + 1)(2x^3 -1) + r = 2x^5 + 2x^3 + x^2 + 1`

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Q8d

Each dividend was divided by another polynomial, resulting in the given quotient and remainder. Find the other polynomial (the divisor).

Dividend: `5x^3 + x^2 +3`

, quotient: `5x^2 -14x + 42`

, remainder: `-124`

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Q9a

Each dividend was divided by another polynomial, resulting in the given quotient and remainder. Find the other polynomial (the divisor).

Dividend: `10x^4 -x^2 + 20x -2`

, quotient: `10x^3 -100x^2 + 999x - 9970`

, remainder: `99 698`

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Q9b

Each dividend was divided by another polynomial, resulting in the given quotient and remainder. Find the other polynomial (the divisor).

Dividend: `x^4 + x^3 - 10x^2 -1`

, quotient: `x^3 - 3x^2 + 2x - 8`

, remainder: `31`

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Q9c

Dividend: `x^3 + x^2 + 7x - 7`

, quotient: `x^2 + 3x +13`

, remainder: `19`

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Q9d

Find a binomial that is a factor of the given polynomial.

`x^3 + 6x^2 -x - 30`

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Q10a

Find a binomial that is a factor of the given polynomial.

`x^4 - 5x^2 +4`

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Q10b

Determine whether each binomial is a factor of the given polynomial.

`x^4 - 5x^2 + 6`

A. `x^2-3`

B. `x+3`

C. `x+1`

D. `x^2+1`

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Q10c

Find a binomial that is a factor of the given polynomial.

`2x^4 -x^3 -4x^2 + 2x + 1`

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Q10d

Determine whether each binomial is a factor of the given polynomial.

`3x + 5, 3x^6+ 5x^5+9x^2+ 17x - 1`

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Q10e

Determine whether each binomial is a factor of the given polynomial.

`5x -1, 5x^4 - x^3 + 10x -10`

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Q10f

The volume of a rectangular box is `(x^3 + 6x^2 + 11x +6)cm^3`

. The
A box is `(x + 3)`

cm long and ` (x + 2) `

cm wide. How high is the box?

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Q11

`8x^3 + 10x^2 - px -5`

is divisible by `2x + 1`

. There is no remainder. Find the value of `p`

.

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Q12a

When `x^6 + x^4 -2x^2 + k`

is divided by `1 + x^2`

, the remainder is ` 5`

. Find the value of `k`

.

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Q12b

The polynomial `x^3 + px^2 -x -2, p\in \mathbb{R}`

, has `x -1`

as a factor. What is the value of `p`

?

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Q13

Let `f(x) =x^n -1`

, where `n`

is an integer and `n \geq 1`

. Name a factor that makes `f(x)`

always divisible by it? Explain.

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Q14

If the divisor of a polynomial, `f(x)`

, is `x - 4`

, then the quotient is
`x^2 + x -6`

and the remainder is 7.

- (a) Write the division statement.
- (b) Rewrite the division statement by factoring the quotient.
- (c) Graph
`f(x)`

using your results in part b).

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Q15

The volume of a cylindrical can is `(4\pi x^3 + 28 \pi x^3 + 65 \pi x + 50\pi)cm^3`

. The can is `(x + 2)`

cm high. What is the radius?

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Q17

Divide.
`(x^4 + x^3y -xy^3 - y^4)\div(x^2 - y^2)`

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Q18a

Divide.
`(x^4 -2 x^3y + 2x^2y^2 - 2xy^3 + y^4)\div(x^2 + y^2)`

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Q18b

Find a factor for `x^3 - y^3`

.

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Q19

Is `f(x) = (x + 5)q(x) + (x + 3)`

, what is the first multiple of `(x + 5)`

that greater than `f(x)`

.

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Q20