Factor completely : when the polynomial has been expressed as a product of a monomial and
one or more prime polynomials
Monomial : constant, a variable or the product of these exponents must be nonnegative integers
*Not Factored Completely* : 7g (5x^2 + 10x + 25)
Should be : 7g (5)(x^2+2x-5) → 35g (x^2 + 2x - 5)
Polynomial : sum or difference of monomials
*Not Factored Completely* : (x-1)[(x-1) + xy(x-1)]
Should be : (x-1)(x-1)(1+xy) → (x-1)^2(1+xy)
Factor Completely : (x-y)+(x-y)(3x+1) → (x-y)(3x+1+1) → (x-y)(3x+2)
Factoring MONIC quadratic trinomials :
MONIC → leading coefficient is 1
Quadratic → 2nd degree
Trinomial → polynomial of 3 terms
ex : q^2 + 3q - 18 → (q+6)(q-3)
Factoring non-monic quadratic trinomials:
Use “splitting the middle”
Ex : 4x^2 - 12x +5
i) Find the product of the leading coefficient and constant : 4(5) = 20
ii) Find the factors of 20 that sum to -12 : -10,-2
iii) Express the linear term using the sum found in (ii) : 4x^2 - 10x - 2x + 5
iv) Group the four terms into pairs : 4x^2 - 10x and -2x + 5
v) Factor each pair : 4x^2 - 10x → 2x(2x-5) and -2x + 5 → -1(2x-5)
Factored form : (2x-1)(2x-5)
Factor : (3x+2)^2 + 8(3x+2) + 12
Substitute 3x + 2 as y
y^2 + 8y + 12 → (y+6)(y+2)
(3x+2+6)(3x+2+2) → (3x+8)(3x+4)
Difference of 2 squares : a^2 - b^2 → (a-b)(a+b)
Factoring hidden quadratics :
x^4 - 7x^2 - 30
Let u = x^2
u^2 - 7u - 30 → (u-10)(u+3) → (x^2-10)(x^2+3)
, Factoring more complex quadratic equations:
1. 25-x^2-4xy-4y^2 → 25-(x+2y)^2 → (5-x-2y)(5+x+2y)
2. 4a^2c^2 - (a^2-b^2+c^2)^2 → (2ac)^2 - (a^2-b^2+c^2)^2
To find the vertex of a quadratic equation in general form (y=ax^2+bx+c):
x=(-b/2a) → Axis of symmetry, substitute this into original equation to find y coordinate of vertex
If a parabola y=ax^2+bx+c intersects the x-axis, then y=0. Therefore, 0=ax^2+bx+c, and the
points of intersection (x-int) are the Real roots of 0=ax^2+bx+c
Powers of i :
i = square root of -1
i^2 = -1
i^3 = - square root of -1
i^4 = 1
CYCLE OF 4
The number i, called the imaginary unit (or imaginary number), is a solution to the equation
x^2+1=0
Pure imaginary numbers: any number that can be expressed in the form bi where b is real and
not equal to 0
𝑎 𝑎
= 𝑏
𝑖𝑓 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑏𝑜𝑡ℎ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒, 𝑜𝑟 𝑖𝑓 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑚 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
𝑏
𝑎𝑏 = 𝑎 × 𝑏 𝑖𝑓 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑎 𝑜𝑟 𝑏 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
Product of 2 pure imaginary numbers is real
The set of complex numbers: the set of pure imaginary numbers bi together with the set of real
numbers
A complex number is of the form a+bi, where a and b are real numbers
(a+bi)(c+di) → ac + adi + cbi - bd → ac - bd is real
When the real part equals 0, you get a pure imaginary number
If r1 and r2 are roots (leading coefficient is 1) → (x-r1)(x-r2) = 0
x^2 - (r1+r2)x + r1r2 = 0
R1+r2 → sum of roots, r1r2 → product of roots
-b/a will be sum of roots, c/a will be product of roots
one or more prime polynomials
Monomial : constant, a variable or the product of these exponents must be nonnegative integers
*Not Factored Completely* : 7g (5x^2 + 10x + 25)
Should be : 7g (5)(x^2+2x-5) → 35g (x^2 + 2x - 5)
Polynomial : sum or difference of monomials
*Not Factored Completely* : (x-1)[(x-1) + xy(x-1)]
Should be : (x-1)(x-1)(1+xy) → (x-1)^2(1+xy)
Factor Completely : (x-y)+(x-y)(3x+1) → (x-y)(3x+1+1) → (x-y)(3x+2)
Factoring MONIC quadratic trinomials :
MONIC → leading coefficient is 1
Quadratic → 2nd degree
Trinomial → polynomial of 3 terms
ex : q^2 + 3q - 18 → (q+6)(q-3)
Factoring non-monic quadratic trinomials:
Use “splitting the middle”
Ex : 4x^2 - 12x +5
i) Find the product of the leading coefficient and constant : 4(5) = 20
ii) Find the factors of 20 that sum to -12 : -10,-2
iii) Express the linear term using the sum found in (ii) : 4x^2 - 10x - 2x + 5
iv) Group the four terms into pairs : 4x^2 - 10x and -2x + 5
v) Factor each pair : 4x^2 - 10x → 2x(2x-5) and -2x + 5 → -1(2x-5)
Factored form : (2x-1)(2x-5)
Factor : (3x+2)^2 + 8(3x+2) + 12
Substitute 3x + 2 as y
y^2 + 8y + 12 → (y+6)(y+2)
(3x+2+6)(3x+2+2) → (3x+8)(3x+4)
Difference of 2 squares : a^2 - b^2 → (a-b)(a+b)
Factoring hidden quadratics :
x^4 - 7x^2 - 30
Let u = x^2
u^2 - 7u - 30 → (u-10)(u+3) → (x^2-10)(x^2+3)
, Factoring more complex quadratic equations:
1. 25-x^2-4xy-4y^2 → 25-(x+2y)^2 → (5-x-2y)(5+x+2y)
2. 4a^2c^2 - (a^2-b^2+c^2)^2 → (2ac)^2 - (a^2-b^2+c^2)^2
To find the vertex of a quadratic equation in general form (y=ax^2+bx+c):
x=(-b/2a) → Axis of symmetry, substitute this into original equation to find y coordinate of vertex
If a parabola y=ax^2+bx+c intersects the x-axis, then y=0. Therefore, 0=ax^2+bx+c, and the
points of intersection (x-int) are the Real roots of 0=ax^2+bx+c
Powers of i :
i = square root of -1
i^2 = -1
i^3 = - square root of -1
i^4 = 1
CYCLE OF 4
The number i, called the imaginary unit (or imaginary number), is a solution to the equation
x^2+1=0
Pure imaginary numbers: any number that can be expressed in the form bi where b is real and
not equal to 0
𝑎 𝑎
= 𝑏
𝑖𝑓 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑏𝑜𝑡ℎ 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒, 𝑜𝑟 𝑖𝑓 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑚 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
𝑏
𝑎𝑏 = 𝑎 × 𝑏 𝑖𝑓 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑎 𝑜𝑟 𝑏 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒
Product of 2 pure imaginary numbers is real
The set of complex numbers: the set of pure imaginary numbers bi together with the set of real
numbers
A complex number is of the form a+bi, where a and b are real numbers
(a+bi)(c+di) → ac + adi + cbi - bd → ac - bd is real
When the real part equals 0, you get a pure imaginary number
If r1 and r2 are roots (leading coefficient is 1) → (x-r1)(x-r2) = 0
x^2 - (r1+r2)x + r1r2 = 0
R1+r2 → sum of roots, r1r2 → product of roots
-b/a will be sum of roots, c/a will be product of roots
