Factoring & Properties of Real Numbers Flashcards
Distributive property
A(2 + 3)
Property of opposite
A + (-a) = 0
Identity property of addition
A + 0 = A
Substitution
(12+12) can be changed to 24
Transitive
If a = b and b = c then a = c
Symmetric
If a = b then b = a
Reflexive
A = a
Associative
(A+b)+c =a+(b+c)
Commutative property
A+b = b+a
Natural
Counting numbers whole not including zero.
Whole numbers
Positive whole numbers including zero
Integers
Positive, negative, whole, zero
Irrational
Can’t be represented as a fraction. Can’t be repeating or terminated
Rational number
Written as fraction (repeats and terminates)
Real number
Not imaginary
Factoring by grouping
x(x+2) + 3(x+2) = (x+3)(x+2)
Zero product Property with factoring
x^2 + 6x = 0
x^2 + 6x = 0
x(x+6) = 0
x = 0 or x = -6
Difference of two squares
a^2 - b^2 = (a+b)(a-b)
Sum of Cubes
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Difference of Cubes
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
(a + b)^2
Perfect Square trinomial (positive)
= a^2 + 2ab + b^2
(a-b)^2
Perfect square trinomial (negative)
= a^2 - 2ab + b^2
sum of two squares
a^2 + b^2 = PRIME
Finding the middle term of 16x^2 + kx + 81
The square root of a (coefficient of x^2) and c (constant) times two
16x^2 + kx + 81
2(4x)(9) = 72x so k = 72
Leading coefficient is 1 (x^2 + bx + c)
Divide the middle term into two numbers that multiply to a * c and add up to the middle coefficient.
x^2 + 8x + 15
1 * 15 = 15
Two numbers that multiply to 15 and add up to 8 are 3 and five
(x + 5)(x +3)
FOLLOW THE SYMBOL OF THE SECOND TERM
Why is x^2 - 18x - 81 PRIME?
Because the C (-81) in a perfect square trinomial can’t be negative.
Why ix x^2 + 81 PRIME?
It can be rewritting as x^2 = 9^2, which is always prime because a^2 + b^2 is always prime.
How can x^2 - 10xy + 25y^2 be simplified?
2(x)(5y) = 10xy, so this is a perfect square trinomial. This means that we can use the formula
(square root of x +/- square root of c)^2
The +/- symbol is determined by the second term. In this case it is being subtracted, so the simplified equation is (x - 5y)^2
How can 81m^2 + 18m + 1 be simplified.
18m is 2(9m)(1) so this is a perfect square trinomial. The middle term is being added. This can be written as (9m + 1)^2
Evaluate (10.1)^2 - 4.2 * 10.1 + (2.1)^2
Use scrap paper.
(10.1-2.1)^2
This is a perfect square trinomial because the middle term is equal to 2(2.1)(10.1) so this is a perfect square. The formula (square root of a +/- square root of c)^2 can be used.
How would you go about doing (3x +1)^2?
(3x + 1)(3x + 1)
Don’t be lazy and distribute the exponent bc that is WRONGNGNGN
Formula for a^2 + b^2
(A + b)(a - b)
(a^3 + b^3)
(A + b)(a^2 - 2b + b^2)
A cubed plus b cubed
(a + b)(a^2 - ab + b^2):
For a trinomial to be a perfect square it must follow this rule…
c = (b/2)^2
