Advanced II Number Sense such as Radicals, Factoring (10) Flashcards
What is a cubed root?
If given a volume, the cubed root of that volume will equal the side length of a cube with that volume:
label the parts of a radical
Notice that the index is the number on top of the radical sign, but that if there is no number then the index is 2.
How can you use factoring to go from an entire radical to a mixed radical?
In order to bring the numbers outside of the radical sign, we need to factor to find things that have the same exponent as the index. When they are taken out from under the radical sign, they lose their exponent since the index is cancelling out that exponent.
What is a rational number?
A rational number is any number that can be written as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers, and the denominator is not equal to zero. In other words, a rational number can be expressed as p/q, where p and q are both integers and q ≠ 0
What is an irrational number?
An irrational number is a real number that cannot be expressed as a ratio of integers; for example, √2 is an irrational number. We cannot express any irrational number in the form of a ratio, such as p/q, where p and q are integers, q≠0
What does it mean to have a negative exponent?
Negative exponents are a way of telling us to use the reciprocal.
What does it mean to have a fractional exponent? (also known as a rational exponent)
The top number of the fractional exponent is just the exponent as you are used to. The bottom number of the fractional exponent tells you the index of the radical sign.
What is the area model for multiplying two polynomials together?
When you multiply, you can choose to multiply in parts such as (10 + 2) ( 30+4) can be thought of as 10(30) + 10(4) + 2(30) + 2(4) = 138
and you can prove this works by using the area model (distributive property and FOIL –> multiply first terms, then outside terms, then inside terms, then last terms, are examples of the area model in very specific situations).
With polynomials you can use this area model as well. The image shows two binomials multiplied together that happen to form a trinomial. If you do not like this method, you can use FOIL for this case as well. FOIL only works for a binomial multiplied by a binomial. The area model works for multiplying anything.
What does it mean to factor?
How can you apply this knowledge to factor out the GCF of a polynomial where each term is divisible by your factor?
Try factoring 12, and then try with 12x+24y + 36 to see the similarities and differences of factoring a polynomial vs. factoring a whole number.
Factor as a verb means to find what the expression is divisible by (this is a factor), and to express that number as a product of that factor. To factor completely means to continue to factor until there are no more factors that can be found.
For example you can say that 12 is divisible by 2, and thus 2 is a factor. Then you could write 12 = 2(6).
To factor completely you may realize that 6 is also divisible by 2 and thus you could say 2 is a factor again. So then 12 = 2(2)(3).
So the factors of 12 are 2, 2 and 3
For polynomials you could do the following:
Given 12x+24y + 36
You could recognize that the entire expression is divisible by 12 (or maybe you notice 2 first and that would work too, you would just have more steps):
So 12 is a factor and the expression is equal to 12(x+2y+3)
When we factor polynomials we do not bother to show the prime factors of the number 12 since it doesn’t serve our purpose. Here we commonly say that 12 is a factor and x+2y+3 is another factor. This is considered to be factored completely.
Factoring can be seen as reversing distributive property here. (factoring undoes multiplication that was done before, so distributive property multiplied and thus we can undo the distributive property by factoring or dividing).
What is the decomposition technique to factor a trinomial?
Try with this one:
8y^2 + 22yz + 15z^2
Factoring by decomposition attempts to undo the multiplication of two binomials. So decomposition undoes FOIL expanding.
However it is not as easy to factor by decomposition since it is tricky to split up the middle term of the trinomial. Follow the steps in the image to get a good sense of what you will need to know.
There are shortcuts in specific cases for this type of factoring, but if you would rather know a method that works in all cases then start here. It will help you to factor any trinomial that is factorable and also will help you to prove if it is factorable or not. Other methods do not offer this heightened insight.
Check your work by using the area model or FOIL to multiply the factors back together (expand) and then simplify. It should return you to the original question again.
