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
example Find the value of k if the remainder of P(x) = 3x^2 -2kx + 5 when divided by x - 2 is 4.
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 - cis a factor ofP(x)thenP(c) = 0
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.
example For P(x)
P(0) = 0P(1) = 0P(3) =0
and deg(P) =3 and P(-1) = 10, find P(x).
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.
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.
Which of the following functions are divisible by x- 1?
f(x) = x^4 - 15x^3 + 2x^2 + 12x - 10
Which of the following functions are divisible by x- 1?
f(x) = 5x^3 - 4x^2 + 3x -4
Is the function divisible by x- 1?
h(x) = x^4 - 7x^3 + 2x^2 + 9x
Is the functions are divisible by x- 1?
j(x) = x^3-1
Determine all the factors of the function f(x) = x^3 + 2x^2 - 5x - 6
State the remainder when x + 2 is divided into polynomial x^2 + 7x + 9
State the remainder when x + 2 is divided into polynomial 6x^3 + 19x^2 + 11x -11
State the remainder when x + 2 is divided into polynomial x^4 - 5x^2 + 4
State the remainder when x + 2 is divided into polynomial x^4 - 2x^3 -11x^2 +10x -2
State the remainder when x + 2 is divided into polynomial x^3 + 3x^2 -10x +6
State the remainder when x + 2 is divided into polynomial 4x^4 +12x^3 - 13x^2 -33x + 18
Determine whether 2x - 5 is a factor of polynomial
x^3 - 3x^2 -10x + 24
Determine whether 2x - 5 is a factor of polynomial
4x^3 + 12x^2 -x - 15
Determine whether 2x - 5 is a factor of polynomial
x^4 + 8x^3 + 4x^2 -48x
Determine whether 2x - 5 is a factor of polynomial
4x^4 + 7x^3 - 80x^2 - 21x + 270
Factor each polynomial using the factor theorem.
x^3 - 3x^2 -10x + 24
Factor each polynomial using the factor theorem.
4x^3 + 12x^2 -x - 15
Factor each polynomial using the factor theorem.
x^4 + 8x^3 + 4x^2 -48x
Factor each polynomial using the factor theorem.
4x^4 + 7x^3 - 80x^2 - 21x + 270
Factor each polynomial using the factor theorem.
x^5 - 5x^4 - 7x^3 + 29x^2 + 30x
Factor each polynomial using the factor theorem.
x^4 + 2x^3 -23x^2 -24x + 144
Factor fully.
f(x) =x^3 + 9x^2 + 8x - 60
Factor fully.
f(x) = x^3 - 7x - 6
Factor fully.
f(x) = x^4 - 5x^2 + 4
Factor fully.
f(x) = x^4 + 3x^3-38x^2 +24x + 64
Factor fully.
f(x) =x^3 -x^2 + x -1
Factor fully.
f(x) = x^5 -x^4 + 2x^3 -2x^2 + x -1
Which of the following can be the graph of below?
f(x) = x^4 - 5x^2 + 4
Which of the following can be the graph of below?
f(x) = x^4 + 3x^3-38x^2 +24x + 64
Use the factored form of f(x) to sketch the graph of each functions.
f(x) =x^3 -x^2 + x -1
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
The polynomial 12x^3 + kx^2 - x - 6 has 2x - 1 as one of its factors. Determine the value of k.
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.
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.
The function f(x) = ax^3 - x^2 + bx - 24 has three factors. Two of these factors are x - 2and x + 4. Determine the values of a and b, and then determine the other factor.
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.
Show that x -a is a factor of x^4 - a^4.
Use the factor theorem to prove that x^2 -x - 2 is a factor of x^3 - 6x^2 + 3x + 10.
Prove that x + a is a factor of (x + a)^5 + (x + c)^5 + (a-c)^5.