C7 Distribution

Question 1

distributing items is the act of distributing them out equally. Algebraic distribution means to…?

Answer 1

multiply each term inside the parentheses by another term that is outside the parentheses. Each term gets multiplied by the same amount.

Question 2

rule for distribution of a term over several others

Answer 2

multiply each individual term in a grouped series of terms by a value outside of the grouping

Question 3

what is a term

Answer 3

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

Question 4

you have the option to distribute first or…?

Answer 4

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

Question 5

rule for distributing positives

Answer 5

distributing a positive sign makes no difference in the signs of the terms

Question 6

+(4x + 2y -3z +7) is the same as…?

Answer 6

multiplying the terms by +1

Question 7

distributing negatives

Answer 7

when distributing negative signs each term has a change of sign to its opposite sign

Question 8

distributing -(4x + 2y - 3z + 7) is the same as…?

Answer 8

distributing -1 over each term in the grouping

Question 9

can you reverse the role of distribution and multiply by a term behind the grouping

Answer 9

yes it does not matter because multiplication is also commutative

Question 10

when multipying factors with the same base, you do what?

a^x · a^y = ____?

Answer 10

add the exponents

a^x · a^y​ = a^{x + y}

Question 11

z⁴(z^-4) = ___?

Answer 11

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

Question 12

formula for changing a negative exponent to a fraction?

a^-n = ___?

Answer 12

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

Question 13

when multiplying factors with the same base the exponents get added together. What is the rule if the exponent is a fraction?

Answer 13

the fractions must have the same denominator to be added (the rules don’t change just because the fractions are exponents)

Question 14

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 = ?

Answer 14

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 = x^1/2y^1/2

Question 15

distribution is easier when you have radicals in the problem if you first change everything to ____ _____.

Answer 15

distribution is easier when you have radicals in the problem if you first change everything to fraction exponents.

Question 16

the rule for raising a product in parentheses to a power is to…?

Answer 16

multiply each power in the parentheses by the outside power

(x⁴)² = x⁸

Question 17

the word polynomial comes from poly meaning “____” and nomen meaning “_____” or “_____”

Answer 17

the word polynomial comes from poly meaning “many” and nomen meaning “name“ or “designation”

Question 18

a pol·y·no·mi·al is an algebraic expression with…?

Answer 18

one or more terms in it

Question 19

what is a polynomial with one term called?

Answer 19

a mo·no·mi·al

Question 20

a polynomial with two terms is called?

Answer 20

a bi·no·mi·al

Question 21

if a polynomial has three terms it is called?

Answer 21

tri·no·mi·al

Question 22

distributing binomials: what are the steps

Answer 22

1. break the binomial into two terms
2. distribute each term of the binomial over the other factor
3. do the distributions you’ve created
4. Simply and combine any like terms.

Question 23

distributing trinomials: what are the steps

Answer 23

1. multiply by distributing the first factor over the second: (x + y + 2)(2y + 3)
2. distribute each term of the trinomial by multipling them times the second factor: x(2y + 3) + y(2y + 3) + 2(2y +3)
3. do the three distributions: 2xy + 3x + 2y² + 3y + 4y + 6
4. simply and combine like terms: 2xy + 3x + 2y² + 7y + 6

Question 24

multiplying one polynomial by another polynomial: the rule for basically all

Answer 24

multiply each each term in the first factor times each of the terms in the second factor:

1. Seperate the terms in the first factor from one another.
2. Multiply each term in the first factor times the second factor
3. Distribute and do the mulitplications
4. Combine like terms

Question 25

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) = _____

Answer 25

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

Question 26

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) = ______

Answer 26

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

Question 27

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) = ______

Answer 27

(x - 4) (x + 4) = x² - 16

Question 28

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) = ____

Answer 28

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

Question 29

a numbers opposite is what?

Answer 29

that same number with a different sign in front

Question 30

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.

Answer 30

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.

Question 31

what does it mean to cube something in algebra?

Answer 31

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