Polynomial Functions
Add
(4x3+3x2−2x+7) + (5x3−x2+6x−4)
Step 1: Group like terms
(4x3+5x3)+(3x2-x2)+(-2x+6x)+(7-4)
Step 2: Add the like terms
=9x3 + 2x2 + 4x + 3
Writing a formula for the input of a function
f(x)=2x+7
Step 1: Change f(x) with y
y=2x+7
Step 2: Solve for x [subtract 7 and divide by 2 from both sides]
x=y−7/2
Is this function exponential growth or decay ?
f(x) e-0.6x
Decay because 0.6 is less than 1.
Add with Like Denominators
x+2/x-1 + 3/x-1
Step 1: Add the numerators
(x+2)+3 = x+5
=x+5/x-1
Identify if it's an Arithmetic Sequence
7,11,15,19..
Step 1: Identify the common difference
11-7=4
15-11=4
19-15-4 so d=4
Yes, it's an arithmetic sequence
Subtract
(5x3+2x2−3x+7) − (2x3−4x2+x−5)
Step 1: Distribute with correct sign values
5x3+2x2−3x+7−2x3+4x2−x+5
Step 2 : Combine like terms
=3x3 + 6x2 − 4x + 12
Find the inverse of a Linear Function
f(x)= 2x+5
Step 1: Make f(x) a y
y=2x+5
Step 2: Switch y with x
x=2y+5
Step 3: Solve for y (Subtract both sides by 5 and divide by 2)
x−5/2 =y
Step 4 : Rewrite as an inverse form
f-1(x)=2x+5
A scientist has 100 grams of a radioactive substance. Every 5 years, the substance decays by half.
How much of the substance will remain after 15 years?
Use this equation A(t)=A0⋅(r)t
Step 1: Plug in the values
A(15)=100⋅(0.5)3
A(15)=100⋅0.125=12.5
12.5 grams remains after 15 years.
Subtracting with Like Denominators
5x+4/x2-9 - 2x-1/x2-9
Step 1 : Subtract the numerators
(5x+4) - (2x-1)
Step 2 : Combine the terms in the numerator
=3x+5
Step 3: Now plug in the new numerator with the denominator
=3x+5/x2-9
On a graph is Arithmetic Linear or Exponential?
Arithmetic is Linear because it increases by a constant difference.
Multiply
(2x+3)(x2−x+4)
Step 1 : Use Distributive Property
2x3 − 2x2 + 8x + 3x2 − 3x + 12
Step 2 : Combine Like Terms
2x3 + x2 + 5x + 12
Approximate to the nearest thousand
log32.6
Step 1: Change of Base
log32.6=loge2.6/loge3
=In2.6/In3
= 0.8697
Step 2: Round
0.870
Adding with Rational Functions Unlike Denominators
2/x2-x + 3/x2-1
Step 1: Factor Both Denominators
2/x(x−1)+3/(x−1)(x+1)
Step 2: Write both with their least common denominator (x-1)(x+1)
2(x+1)/(x-1)(x+1)
and
3x/x(x−1)(x+1)
Step 3: Add the numerators and plug it in with the denominators
5x+2/x(x-1)(x+1)
Write a rule for the ntn term
12,17,22,27
Step 1: Identify the common difference
17−12=5
22−17=5
27−22=5 so d=5
Step 2: Use this formula an=a+(n−1)d [where d=5 and a=12]
an=12+(n−1)5
Step 3: Simplify
an=12+5n−5=5n+7
an=5n+7
Factor this Polynomial by Grouping
x3 + 3x2 + 2x + 6
Step 1: Pair the terms that have a common monomial factor
x2(x+3)+2(x+3)
= (x2+2)(x+3)
Solve
7log11(a-8)=14
Step 1: Solve for a (divide by 7, subtract by 8)
log11(a-8)=2
112=a-8
121=a-8
129=a
Multiplying Rational Expressions
(2x/x2-9)(x2+3x/4x)
Step 1: Factor and multiply across
2x2(x+3)/4x(x-3)(x+3)
Step 2: Cancel common factors
2x/4(x-3)
Step 3 : Simplify
x/2(x-3)
Does this series have a sum ?
OO
5(-2.2)n-1
n=1
There is no sum because -2.2 doesn't range from -1 to 1.
Solve this Polynomial Equation by Factoring
x2 − 7x + 12 = 0
Step 1: Factor (find two numbers that multiply to 12 and add to -7)
(x−3)(x−4)
x=3 x=4
Solve
5 x 7-3x+9=22
Step 1: Solve for X (divide by 5..)
In (7-3x)=In(13/5)
-3xIn(7)=In(13/5)
X=In(13/5)/-3In(7)
Solve a Rational Equation by Cross Multiplying
2x+3/5 = x-1/3
Step 1: Cross Multiply
3(2x+3)=5(x−1)
Step 2 : Distribute
6x+9=5x−5
Step 3: Solve for x ( Subtract 5x and 9 from both sides)
x=-14
A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 80% of the height from which it fell. What is the total distance the ball travels if it bounces infinitely many times?
Step 1: Use the formula S= a/1-r
S= 8/1−0.8 = 8/0.2= 40
Step 2 : Add the first drop
10+40=50
Total Distance is 50 meters