Question

A rectangle has a width of 2x-3 and a length of 3x+1
a) Write its area as a simplified polynomial
b) Write expressions for the dimensions if the width is doubled and the length is increased by 2
c) Write the new area as a simplified polynomial

Answer 1

Where are you having issues with this problem?

Answer 2

@LifeEngineer I don't know what to do

Answer 3

What is the formula for the area of a triangle

Answer 4

*rectangle

Answer 5

base times height

Answer 6

Area = Length times Width

Answer 7

So multiply the length, by the width

Answer 8

Whoops! Yea, length times width is what I meant to say

Answer 9

(2x-3)(3x+1)
like that?

Answer 10

yes

now foil this

Answer 11

Length = 3x+1
Width = 2x-3

Area = (2x-3)(3x+1)

FOIL

First, Outer, Inner, Last

First = \[2x \times 3x\]

First = 6x

Outer = \[2x \times 1 \]

Outer = 2x

Inner = \[-3 \times 3x\]

Inner = -9x

Last = -3 x 1

Last = -3

Answer 12

So now we add them all up

First is actually 6x^2

Answer 13

\[6x^2 + 2x - 9x - 3\]

What do we get?

Answer 14

(2x-3)(3x+1)
=6x^2+2x-9-3
=6x^2+2x-12

Answer 15

thatʻs not right

Answer 16

Itʻs -9x , not -9

Answer 17

(2x-3)(3x+1)
=6x^2+2x-9x-3
=6x^2-7x-3

Answer 18

Correct

Answer 19

We have solved a)

Answer 20

Is that the area?

Answer 21

Yes

Answer 22

Does that mean that I have to put cm^2 at the end of it?

Answer 23

If that is the unit of measure, then yes

Answer 24

Now

b) Write expressions for the dimensions if the width is doubled and the length is increased by 2

Width = 2x-3
Length = 3x+1

Doubled is multiplication

We basically multiply 2x-3 by itself

This is represented by...

(2x - 3)^2

which is basically

(2x - 3)(2x - 3) So we foil again

Answer 25

Can you try do that one?

Answer 26

(2x-3)(2x-3)
=4x^2-6x-6x+9
=4x^2-12x+9

Answer 27

Correct, now that is the doubled form of 2x - 3

Answer 28

Now we just increase 3x+1 by 2

Answer 29

Do you know what "increased by" means?

Answer 30

(3x+1)(3x+1)
=9x^2+3x+3x+1
=9x^2+6x+1

Answer 31

No...that is doubling it

Answer 32

We donʻt want to multiply it by itself, we want to do a different operation with the number 2

Answer 33

What does increased by 2 mean?
To the power of two?

Answer 34

(3x+1)^2
like that?

Answer 35

no

Answer 36

Let says I have a laser gun. It is currently at power level two. If I increase the power level by 2, what level will it be at?

Answer 37

4 ?

Answer 38

Yes, but what mathematical operation would you do to get there?

Answer 39

addition?

Answer 40

Yes

Answer 41

So 3x + 1 increased by 2

Answer 42

3x+1 turns into.... 5x+3

Answer 43

How did you get that

Answer 44

3x +2 = 5x
1 +2 = 3

Answer 45

No...

You canʻt do 3x + 2

Do you know about like terms?

Answer 46

2 does not have the variable x, thus it is not like terms with 3x, and cannot be added or subtracted from each other

Answer 47

They can, however, multiply and divide each other

Answer 48

Yes, I'm familiar with like terms. I'm just confused.

Answer 49

So

3x + 1 increased by 2

What do we get? ^-^

Answer 50

3x times 2 = 6x
1 times 2 = 2

Answer 51

6x+2

Answer 52

noooo

We are doing addition!

Answer 53

But you said:
"2 does not have the variable x, thus it is not like terms with 3x, and cannot be added or subtracted from each other"

Answer 54

I thought we didn't have to do addition

Answer 55

Exactly

BUT

increased by = +

So

3x + 1 + 2

Answer 56

We donʻt add the 2 to 3x.

Answer 57

So would 3x + 1 + 2 be my final answer or would it be 3x+3 ?

Answer 58

Yes

Answer 59

Okay and what's the new area?

Answer 60

c) Write the new area as a simplified polynomial

Our doubled width = 4x^2-12x+9

Our new length = 3x + 3

Answer 61

So we have to multiply those two new dimensions

Answer 62

(4x^2-12x+9)(3x+3)
Do I have to foil this?

Answer 63

We cannot FOIL this

(4x^2-12x+9)(3x+3)

First we start by pulling out like factors

3x + 3 = 3 • (x + 1)

or 3(x + 1)

Answer 64

Next, we factor by splitting the middle term

Answer 65

Factoring 4x2 - 12x + 9

The first term is, 4x2 its coefficient is 4 .
The middle term is, -12x its coefficient is -12 .
The last term, "the constant", is +9

Multiply the coefficient of the first term by the constant 4 • 9 = 36

Find two factors of 36 whose sum equals the coefficient of the middle term, which is -12 .

-36 + -1 = -37
-18 + -2 = -20
-12 + -3 = -15
-9 + -4 = -13
-6 + -6 = -12

Answer 66

Rewrite the polynomial splitting the middle term using the two factors found in the above post

-6

4x2 - 6x - 6x - 9

Add up the first 2 terms, pulling out like factors

2x • (2x-3)

Add up the last 2 terms, pulling out common factors

3 • (2x-3)

Add up the four terms of previous step

(2x-3) • (2x-3)

Which is the desired factorization.

We multiply (2x-3)(2x-3)

We get = 3 • (2x - 3)2 • (x + 1)

Answer 67

Correction: 3(2x - 3)^2 • (x + 1)

Answer 68

Simplified that is 12x^3 -24x^2 - 9x + 27

Answer 69

That is your new area