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Lecture on Rational Zero Theorem
4 Videos

Lecture on Remainder and Factor Theorem
8 Videos

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Introduction to Remainder Theorem

*example* Find the remainder when `P(x) = 3x^5 + 4x^4 -4x^3 + 7x + 3`

is divided by `x + 2`

. Use the remainder theorem. Also, find it using synthetic division. Which solution is simpler

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ex1 of Remainder Theorem

*example* Find the value of `k`

if the remainder of `P(x) = 3x^2 -2kx + 5`

when divided by `x - 2`

is `4`

.

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1.04mins

ex3 Finding k when remainder is given

**Factor Theorem**

Every Polynomial can be broken into the following format.

`P(x) = D(x) Q(x) + R(x)`

Then we can say

- if
`x - c`

is a factor of`P(x)`

then`P(c) = 0`

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Introduction to Factor Theorem

*example* Find the value of `a`

and `b`

when `P(x) =ax^3 -(b -1)x^2 + x - 2`

is divisible by `x^2 -2x - 3`

.

**Key Ideas**

`x^2 -2x - 3`

has two factors `(x -3)`

and `(x + 1)`

.

Use Remainder theorem with the two factors.

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Using Factor Theorem to find the missing coefficients in the polynomial

*example* For `P(x)`

`P(0) = 0`

`P(1) = 0`

`P(3) =0`

and `deg(P) =3`

and `P(-1) = 10`

, find `P(x)`

.

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Ex2 Finding Polynomial given root info

*example* Find the remainder when `P(x) = 3x^3 -x^2 + x - 3`

is divided by `D(x) = x^2 - x - 2`

, using the remainder theorem.

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Given the remainder of a polynomial with two polynomial of degree one linear polynomial finding

Solutions
40 Videos

Given `f(x) = x^4 + 5x^3 + 3x^2 - 7x + 10`

, determine the remainder when f(x) is divided by each of the following binomials, without dividing:

(i) `x - 2`

(ii) `x + 4`

(iii) `x-1`

Are any of the binomials in part (a) factors of f(x)? Explain.

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Q1

Which of the following functions are divisible by `x- 1`

?

`f(x) = x^4 - 15x^3 + 2x^2 + 12x - 10`

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Q2a

Which of the following functions are divisible by `x- 1`

?

`f(x) = 5x^3 - 4x^2 + 3x -4`

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Q2b

Is the function divisible by `x- 1`

?

`h(x) = x^4 - 7x^3 + 2x^2 + 9x`

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0.20mins

Q2c

Is the functions are divisible by `x- 1`

?

`j(x) = x^3-1`

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Q2d

Determine all the factors of the function `f(x) = x^3 + 2x^2 - 5x - 6`

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Q3

State the remainder when `x + 2`

is divided into polynomial `x^2 + 7x + 9`

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Q4a

State the remainder when `x + 2`

is divided into polynomial `6x^3 + 19x^2 + 11x -11`

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0.42mins

Q4b

State the remainder when `x + 2`

is divided into polynomial `x^4 - 5x^2 + 4`

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Q4c

State the remainder when `x + 2`

is divided into polynomial `x^4 - 2x^3 -11x^2 +10x -2`

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0.42mins

Q4d

State the remainder when `x + 2`

is divided into polynomial `x^3 + 3x^2 -10x +6`

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Q4e

State the remainder when `x + 2`

is divided into polynomial `4x^4 +12x^3 - 13x^2 -33x + 18`

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Q4f

Determine whether `2x - 5`

is a factor of polynomial

`x^3 - 3x^2 -10x + 24`

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Q5a

Determine whether `2x - 5`

is a factor of polynomial

`4x^3 + 12x^2 -x - 15`

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Q5b

Determine whether `2x - 5`

is a factor of polynomial

`x^4 + 8x^3 + 4x^2 -48x`

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Q5c

Determine whether `2x - 5`

is a factor of polynomial

`4x^4 + 7x^3 - 80x^2 - 21x + 270`

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2.02mins

Q5d

Factor each polynomial using the factor theorem.

`x^3 - 3x^2 -10x + 24`

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Q6a

Factor each polynomial using the factor theorem.

`4x^3 + 12x^2 -x - 15`

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1.08mins

Q6b

Factor each polynomial using the factor theorem.

`x^4 + 8x^3 + 4x^2 -48x`

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1.01mins

Q6c

Factor each polynomial using the factor theorem.

`4x^4 + 7x^3 - 80x^2 - 21x + 270`

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2.09mins

Q6d

Factor each polynomial using the factor theorem.

`x^5 - 5x^4 - 7x^3 + 29x^2 + 30x`

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Q6e

Factor each polynomial using the factor theorem.

`x^4 + 2x^3 -23x^2 -24x + 144`

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Q6f

Factor fully.

`f(x) =x^3 + 9x^2 + 8x - 60`

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Q7a

Factor fully.

`f(x) = x^3 - 7x - 6`

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Q7b

Factor fully.

`f(x) = x^4 - 5x^2 + 4`

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Q7c

Factor fully.

`f(x) = x^4 + 3x^3-38x^2 +24x + 64`

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Q7d

Factor fully.

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

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Q7e

Factor fully.

`f(x) = x^5 -x^4 + 2x^3 -2x^2 + x -1`

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Q7f

Which of the following can be the graph of below?

`f(x) = x^4 - 5x^2 + 4`

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Q8c

Which of the following can be the graph of below?

`f(x) = x^4 + 3x^3-38x^2 +24x + 64`

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Q8d

Use the factored form of `f(x)`

to sketch the graph of each functions.

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

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Q8e

Use the factored form of `f(x)`

to sketch the graph of each functions.

`f(x) = x^5 -x^4 + 2x^3 -2x^2 + x -1`

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Q8f

The polynomial `12x^3 + kx^2 - x - 6`

has `2x - 1`

as one of its factors. Determine the value of `k`

.

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Q9

When `ax^3 -x^2+2x + b`

is divided by `x -1`

, the remainder is 10. When it is divided by `x - 2`

, the remainder is `51`

. Find `a`

and `b`

.

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Q10

Determine a general rule to help decide whether `x- a`

and `x - a`

are factors of `x^n -a^n`

and `x^n + a^n.`

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Q11

The function `f(x) = ax^3 - x^2 + bx - 24`

has three factors. Two of these factors are `x - 2`

and `x + 4`

. Determine the values of `a`

and `b`

, and then determine the other factor.

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Q12

Consider the function `f(x) = x^3 + 4x^2 + kx - 4`

. The remainder from `f(x) \div (x + 2) `

is twice the remainder from `f(x) \div (x + 2)`

. Determine the value of `k`

.

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Q13

Show that `x -a`

is a factor of ` x^4 - a^4`

.

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Q14

Use the factor theorem to prove that `x^2 -x - 2`

is a factor of `x^3 - 6x^2 + 3x + 10`

.

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Q16

Prove that `x + a`

is a factor of `(x + a)^5 + (x + c)^5 + (a-c)^5`

.

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Q17