Fully Factored Form
Factor and leave in fully factored form:
x2-9
(x+3)(x-3)
What two things influence the end behavior of a graph of a polynomial?
Degree and Leading Coefficient (+ / -)
Is (x-1) a factor of x4-2x3+3x-2?
Yes,
f(1) = 1 - 2 + 3 - 2 = 4 - 4 = 0
Factor and Leave in FULLY Factored Form:
(x2+16)(x2+36)(x2+49)
This polynomial is already in fully factored form!
You can only factor further if it is a difference of two squares, not a sum
What are the real zeros (with multiplicity) of this polynomial?
3x(x+2)2(x-4)(x-3)4
What does multiplicity have to do with a graph of a polynomial?
Zeros:
0 mult 1, -2 mult 2, 4 mult 1, 3 mult 4
multiplicity determines if a graph "touches" or "crosses" the x axis at the zeros
Even multiplicity -- touches
Odd multiplicity -- crosses
What is the remainder when -2x3+7x-2 is divided by x+3?
f(-3) = 31
-2(-3)3 + 7(-3) - 2
Remainder is 31
Factor and leave in fully factored form:
(x2 - 169)(x2 + 10)(x2-16)
What is the maximum number of turning points of this graph?(x-3)(x2+4)
3rd Degree (Cubic)
x3-3x2+4x-12
TPs: degree - 1
2 turning points max
If a 5th degree polynomial has zeros of -4, 2i, and i+3, what are the remaining zeros?
Remaining Zeros: -2i, i-3
Factor and leave in fully factored form:
x3-2x2 -9x+18
x2(x-2)-9(x-2)
(x-2)(x2-9)
(x-2)(x-3)(x+3)
What is the degree of this polynomial?
x4 - x2 - 72
4th degree (quartic)
When a polynomial is written in factored form, degree is the sum of the exponents
When a polynomial is written in standard form, degree is largest exponent
Find a polynomial with the following zeros: -4, 3i. Give the polynomial in standard form
(x+4)(x-3i)(x+3i)
(x+4)(x2+9)
x3+4x2+9x+36
Factor and leave in fully factored form:
x4-12x2-64
(x2-16)(x2+4)
(x-4)(x+4)(x2+4)
What are the end behaviors of all odd degree polynomials with a positive leading coefficient?
What are the end behaviors of all even degree polynomials with a positive leading coefficient?
What would change with a negative leading coefficient?
Odd: right side up, left side down
Even: both sides up
Negative LC means end behaviors flip
(right down left up for odd, both down for even)
Find all zeros of the function given that 5 is a zero:
2x3-12x2-2x+60
SD Using x = 5
Result: 2x2-2x-12
2(x2-x-6)
2(x-3)(x+2)
Remaining Zeros: x = 3,-2