C7 Distribution Flashcards
distributing items is the act of distributing them out equally. Algebraic distribution means to…?
multiply each term inside the parentheses by another term that is outside the parentheses. Each term gets multiplied by the same amount.
rule for distribution of a term over several others
multiply each individual term in a grouped series of terms by a value outside of the grouping
what is a term
it is made up of variable(s) and/or number(s) joined by multiplication and/or division and seperated from another term by addition or subtraction
you have the option to distribute first or…?
perform the operations, addition and subtraction inside the grouping first. Both will give you the same result but one may be preferable depending on the type of problem
rule for distributing positives
distributing a positive sign makes no difference in the signs of the terms
+(4x + 2y -3z +7) is the same as…?
multiplying the terms by +1
distributing negatives
when distributing negative signs each term has a change of sign to its opposite sign
distributing -(4x + 2y - 3z + 7) is the same as…?
distributing -1 over each term in the grouping
can you reverse the role of distribution and multiply by a term behind the grouping
yes it does not matter because multiplication is also commutative
when multipying factors with the same base, you do what?
ax · ay = ____?
add the exponents
ax · ay = ax + y
z4(z-4) = ___?
z0 which equals 1. Do not make the mistake of removing the variable all together. x0 = 1 for any real number except 0.
After thinking this through it is correct because that would be the multiplicative inverse which equals 1
formula for changing a negative exponent to a fraction?
a-n = ___?
place the base and the exponent in the denominator of the fraction, but the exponent loses it’s negative sign. Then place 1 in the numerator
a-n = 1 / an
when multiplying factors with the same base the exponents get added together. What is the rule if the exponent is a fraction?
the fractions must have the same denominator to be added (the rules don’t change just because the fractions are exponents)
radicals can be changed to expressions with ____ as ____. This is handy when you want to combine terms with the same bases and you have some of the bases under the radicals
√ ̅xy = √ ̅x √ ̅y = ?
radicals can be changed expressions with fractions as exponents. This is handy when you want to combine terms with the same bases and you have some of the bases under the radicals
√ ̅xy = √ ̅x √ ̅y = x1/2y1/2
distribution is easier when you have radicals in the problem if you first change everything to ____ _____.
distribution is easier when you have radicals in the problem if you first change everything to fraction exponents.
the rule for raising a product in parentheses to a power is to…?
multiply each power in the parentheses by the outside power
(x4)2 = x8
the word polynomial comes from poly meaning “____” and nomen meaning “_____” or “_____”
the word polynomial comes from poly meaning “many” and nomen meaning “name“ or “designation”
a pol·y·no·mi·al is an algebraic expression with…?
one or more terms in it
what is a polynomial with one term called?
a mo·no·mi·al
a polynomial with two terms is called?
a bi·no·mi·al
if a polynomial has three terms it is called?
tri·no·mi·al
distributing binomials: what are the steps
- break the binomial into two terms
- distribute each term of the binomial over the other factor
- do the distributions you’ve created
- Simply and combine any like terms.
distributing trinomials: what are the steps
- multiply by distributing the first factor over the second: (x + y + 2)(2y + 3)
- distribute each term of the trinomial by multipling them times the second factor: x(2y + 3) + y(2y + 3) + 2(2y +3)
- do the three distributions: 2xy + 3x + 2y2 + 3y + 4y + 6
- simply and combine like terms: 2xy + 3x + 2y2 + 7y + 6
multiplying one polynomial by another polynomial: the rule for basically all
multiply each each term in the first factor times each of the terms in the second factor:
- Seperate the terms in the first factor from one another.
- Multiply each term in the first factor times the second factor
- Distribute and do the mulitplications
- Combine like terms
when the same binomial is multiplied by itself –when each of the first two terms is distributed over the second and same terms – then the resulting ______ contains the _____ of the two terms and ____ their product.
(a + b)2 = (a + b) (a + b) = _____
when the same binomial is multiplied by itself –when each of the first two terms is distributed over the second and same terms – then the resulting trinomial contains the squares of the two terms and twice their product.
(a + b)2 = (a + b) (a + b) = a2 + b2 + 2ab
For me: (3 + 4)2 = (3 + 4) (3 + 4) = 32 + 42 + 2 * 3 * 4 = 9 + 16 + 24 = 49
The sum of any two terms multiplied by their difference equals the ____ of the _____ of these same two terms. For any real numbers a and b: (a + b) ( a - b) = ______
The sum of any two terms multiplied by their difference equals the difference of the squares of these same two terms. For any real numbers a and b: (a + b) ( a - b) = a2 - b2
Notice the middle term just disappears during distribution because a term and its opposite are always in the middle.
(a+b) (a - b) = a(a - b) + b(a - b) = a2 - ab + ab - b2
the sum and difference of the same two terms rule always works so you can use the shortcut to do the special distributions such as (x - 4) (x + 4) = ______
(x - 4) (x + 4) = x2 - 16
The difference or sum of two cubes is equal to the difference or sum of their ____ ____ times a trinomial, which contains the _____ of the cube roots and the opposite of the product of the cube roots. For any real numbers a and b,
(a - b) (a2 + ab + b2) = ____
(a + b) (a2 - ab + b2) = ____
The difference or sum of two cubes is equal to the difference or sum of their cube roots times a trinomial, which contains the squares of the cube roots and the opposite of the product of the cube roots. For any real numbers a and b,
(a - b) (a2 + ab + b2) = a3 - b3
(a + b) (a2 - ab + b2) = a3 + b3
a numbers opposite is what?
that same number with a different sign in front
When multplying two binomials together you can always just FOIL them. You save yourself some work, though, if you can recognize when the terms in the two binomials are the same–except for the sign between them. If they’re the ____ and _____ of the same two numbers, then their product is just the _____ between the squares of the two terms.
When multplying two binomials together you can always just FOIL them. You save yourself some work, though, if you can recognize when the terms in the two binomials are the same–except for the sign between them. If they’re the sum and difference** of the same two numbers, then their product is just the **difference between the squares of the two terms.
what does it mean to cube something in algebra?
it means to multiply a value by itself and then multiply the result by itself
33 = 3 x 3 = 9 and then 9 x 3 = 27
