Remainders
Explain what a remainder is.
A part, number, or quantity that is left over.
What is a polynomial?
An expression consisting of multiple terms
Describe what a factor of a polynomial is.
Breaking up polynomials into simpler terms
Name two types of division
Long division and synthetic division
If (x-2) is a factor of a polynomial, name one possible zero
x=2
Explain the Remainder Theorem.
f(x)=(x-c)*q(x)+r(x), f(c)=(c-c)*q(c)+r, therefore, f(c)=r
What is the quadratic formula?
(-b(+/-)sqrt(b2)-4(a)(c))/(2a)
Explain the Factor Theorem.
When f(c)=0 then x-c is a factor of f(x)
Divide 99 by 11
9
If (x+8) is a factor of a polynomial, what is a possible zero?
x=-8
Use the remainder theorem to find the remainder when 3x3-2x2+x-5 is divided by x+2
(x+2) so f(2),
3(-2)3-2(-2)2+(-2)-5
-24-8-2-5
R= -39
(-6.5, -2)U(2,infinity)
Factor f(x)=x4-7x2+12
f(x)=(x2-4)(x2-3)
f(x)=(x+2)(x-2)(x2-3)
Divide 3x3–x–7 by x–2 using synthetic division
3x2+6x+11 R 15
x=1-3i and x=1+3i
Divide (2x3-7x2+4x-5) by (x-3)
2x2-x+1 R -2
Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicity include 2(multiplicity of 2), 3+i
x4-10x3+38x2-64x+40
Use the Factor Theorem to determine whether x+3 is a factor of x3+2x2-4x-2
f(-3)=1
so not a factor
Find all zeros and write a linear factorization of the function f(x)=x3+x+10
synthetically divide by -2
x2-2x+5
quadratic formula
x=1(+/-)2i
(x+2)(x-1-2i)(x-1+2i)
What are the potential rational zeros of f(x)=2x4-x3-4x2-x-6
+/-1, +/-1/2, +/-2, +/-3, +/-3/2, +/-6
Divide 2x4-3x3+9x2-14x+7 by x2+4
2x2-3x+1 R -2x+3
Using the given zero of 1+3i, find all the zeros of f(x)=x4-2x3+5x2+10x-50
Synthetically divide by 1+3i, synthetically divide by 1-3i
x3-5x2
x(x2-5)
x=+/- 5
Write f(x)=x3-x2-x-2 as a product of linear and irreducible quadratic factors with all real coefficients.
(x-2)(x2+x+1)
Using the given zero, final all of the zeros and write a linear factorization of the function. 1+i is a zero of f(x)=x4-2x3-x2+6x+6
Synthetically divide using 1+i
Synthetically divide using 1-i
x2-3=0
x=sqrt(3), -sqrt(3)
(x-1-i)(x-1+i)(x+sqrt(3))(x-sqrt(3))
Find all the real zeros of the function. Identify each zero as rational or irrational. f(x)=x4-7x2+12
x=+/- 2 and x=+/- sqrt(3) or 3^(1/2)