Fill in the missing expressions to complete the factorizations below?

Question

How do you do these two problems??

64m^3+27=( )(16m^2-12m+9)

v^3-1000=( ) (v^2+10v+100)

Answer 1

These are what is known as sum and difference of cubes problems. What usually goes in the first set of brackets are the cube root of each term, and what usually goes in the second bracket are terms where you (1) square the first, (2) negative product, (3) square the last.

Note that 64 = 4^3 and 27 = 3^3. Therefore,

64m^3 + 27 = (4m + 3) (? + ? + ?)

"Square the first" means you square the first term in the first set of brackets.

(4m + 3)(16m^2 + ? + ?)

"Negative product" means you multiply both terms in the first brackets together, and then negate the result. 4m times 3 is equal to 12m, negated is -12m.

(4m + 3)(16m^2 - 12m + ?)

"Square the last" means you square the last term in the first set of brackets.

(4m + 3)(16m^2 - 12m + 9)

2) v^3 - 1000

Note that 1000 is equal to 10^3. Therefore,

v^3 - 1000 = (v - 10) (v^2 + 10v + 100)

Answer 2

(4m + 3) will acheive desired coeffecients for:

m^3 and m^0. Now let's look at what we get:

64m^3 - 48m^2 + 36m + 48m^2 - 36m + 27

= 64m^3 + 27

So (4m + 3) is correct

2) Again, (v - 10) to get the coefficients for

v^3 and v^0. Check:

v^3 + 10v^2 + 100v - 10v^2 -100v - 1000 =

= v^3 - 1000. So, (v - 10) is correct

Answer 3

Missing parts are 4m + 3 and v-10

Apply the factorization formulae for a^3-b^3 and a^3+b^3