A teacher writes the following product on the board: \qquad (3x^2)(4x) = 12x^3(3x 2 )(4x)=12x 3 left parenthesis, 3, x, squared, right parenthesis, left parenthesis, 4, x, right parenthesis, equals, 12, x, cubed Miles concludes that 3x^23x 2 3, x, squared is a factor of 12x^312x 3 12, x, cubed. Jude concludes that 12x^312x 3 12, x, cubed is divisible by 4x4x4, x.

Both students are right.

Step-by-step explanation:

The product that the teacher wrote on the board is

One of his students called Miles, conclude that

is a factor of

This is very true because from the given product both 3x² and 4x are factors of 12x³.

Another student , Jude also concludes that 12x³ is divsible by 4x.

This is also true because:

Hence both students are correct.

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}

idk lol

step-by-step explanation: