Question

Simplify the expressions below. Write the final product in standard form and show your work to receive full credit.

1) 2x^4(4x^2 + 3x + 1)
2) (4x – 3)(2x^2 – 7x + 1)
3) (x^2 + 4x – 3)(2x^2 + x + 6)

Answer 1

Looks like distributive property should work out...
\(a(b+c) = ab + ac\)
Distributive works with polynomial expressions as well (it just requires more steps to fully simplify)
\(\begin{align} (a+b)(c+d) &= (a+b)c + (a+b)d \\& = ac + bc + ad + bd \end{align} \)

Answer 2

Multiplication of Polynomials is what it is.

Answer 3

Could you explain MY problems to me so I can hurry and understand and turn this in before I fail, please.

Answer 4

Access?

Answer 5

Yeah, and distributive property works for polynomials. :)
Like, for #1:
\( \begin{align}
\color{Green}{2x^4} (4x^2 + 3x + 1) &= \color{green}{2x^4}*4x^2 + \color{green}{2x^4} * 3x + \color{green}{2x^4} * 1 \\
&= 8x^4 * x^2 + 6x^4 * x + 2x^4 \\
&= 8x^{4+2} + 6x^{4+1} + 2x^4 \\
&= 8x^6 + 6x^5 + 2x^4
\end{align} \)

Answer 6

Aright well my instructions are this --> Simplify the expressions below. Write the final product in standard form and show your work to receive full credit. SO, is that simplified above?

Answer 7

Yes. That is simplest form. The degrees are in descending order and there is nothing more that can be done to simplify.

Answer 8

Alright, now I have to write that in Standard form..

Answer 9

I believe that is standard form already...

Answer 10

Um.. I'm confused. It says simplify it and then write the FINAL product in standard form. So, I guess I have to write 8x^6 + 6x^5 + 2x^4 in standard form.

Answer 11

but wouldn't 8x^6 + 6x^5 + 2x^4 already be in Standard form?

Answer 12

Yeah, that's what I was thinking. Standard form, to me, is just all the terms in descending degree / simplest form.

Answer 13

So, what about this one? (4x – 3)(2x^2 – 7x + 1)

Answer 14

As for #2, we'd still use distributive property, just considering the binomial as one "thing" and multiplying it through.
\( \begin{align}
\color{green}{(4x - 3)}(2x^2 - 7x + 1) &= \color{green}{(4x - 3)}* 2x^2 + \color{green}{(4x - 3)}(-7x) + \color{green}{(4x - 3)}*1 \\
&= 2x^2 (4x) + 2x^2 (-3) + (-7x)(4x) + (-7x)(-3) + 4x - 3 \\
&= 8x^3 \underbrace{ - 6x^2 - 28x^2} \underbrace{+21x + 4x} - 3 \\
&= 8x^3 - 34x^2 + 25x - 3 \\
\end{align} \)

Makes sense?

Answer 15

I don't understand this at all...

Answer 16

Do you understand distributive property? That's the basis for what I'm doing in each problem.

All I'm doing is taking one thing multiplied to the other and distributing it through to each term in the other, and then it's just simplifying from there.

Answer 17

Not really.. :/

Answer 18

Sorry if I'm bothering you, I just want to understand it, not just get the answer, you know?

Answer 19

You're not bothering me. Of course, that's completely understandable. I'm just thinking of how to explain it. :)

The basic idea of distributive property is: \(a(b+c) = ab + ac\).
Of course, this will apply if \(a\) is anything being multiplied to any number of terms.
\(a(b+c+d) = ab + ac + ad\)

Say, a=5, b=3, and c=2
5(3+2) = 5(5) = 25 by simplification
5(3+2) = 5(3) + 5(2) = 15 + 10 = 25 by distributive
So it does seem that it works out.

Answer 20

So, looking back at what I was doing earlier, that green term is just the "a" in the distributive property.

Answer 21

The variables confuse me, I think. Because I understand that one completely.

Answer 22

You left didn't you?....

Answer 23

The \(a\), \(b\), and \(c\) just represent anything. That's basically the idea when writing the property with those variables. It just gives us a model
No. It might say I left for some reason, but I won't leave unless you tell me to or I actually "have" to. :P

Answer 24

(x^2 + 4x – 3)(2x^2 + x + 6) I would say that this would be 4x^4 - 3 * 3x^2 + 6.

Answer 25

hmm...
First step is
\( \begin{align}
\color{green}{(x^2 + 4x - 3)}(2x^2 + x + 6) = \color{green}{(x^2 + 4x - 3)}(2x^2) + \color{green}{(x^2 + 4x - 3)}(x) + \color{green}{(x^2 + 4x - 3)}(6)
\end{align} \)
Then we have to distribute one more time.

Answer 26

Oh, that kind of went off the page
\[\begin{align}
&\color{green}{(x^2 + 4x - 3)}(2x^2 + x + 6) \\
&= \color{green}{(x^2 + 4x - 3)}(2x^2) + \color{green}{(x^2 + 4x - 3)}(x) + \color{green}{(x^2 + 4x - 3)}(6)
\end{align}
\]

Answer 27

So, that's it?

Answer 28

We'll have to use distributive one more time with the 2x^2, x, and 6. Then we simplify exponents and add the like terms.

Answer 29

I've been stuck on these 3 questions for 3 hours...

Answer 30

I still don't understand WHEN to distribute and where.

Answer 31

It's a property that applies whenever we multiply two things where one has addition/subtraction. If we have this case anywhere, we can always apply distributive property.

We have: (x^2 + 4x - 3)(2x^2)
The first factor is a polynomial with three terms, which are all connected by addition/subtraction. The second is a single term. This is just like the "a" in the original thing.
So, we can just multiply 2x^2 to each part of the first polynomial.

Answer 32

x^2 (2x^2) + 4x (2x^2) - 3 (2x^2)
= 2x^4 + 8x^3 - 6x^2

Answer 33

So, you're multiplying each part with the last part...?

Answer 34

Yes. Similarly,
(x^2 + 4x - 3)(x) Take the x to each term in the first factor
x^2 (x) + 4x (x) - 3 (x)
= x^3 + 4x^2 - 3x

Answer 35

Now explain it like that without doing it with the equation thing so I can read it better.

Answer 36

ok..

Answer 37

(x2 + 4x – 3)(2x2 + x + 6)

Answer 38

(x^2 + 4x – 3)(2x^2 + x + 6)

Answer 39

okay (i just use the equation thing to color the terms I'm distributing, I guess it didn't help then. :P)

(x^2 + 4x - 3)(6) multiply 6 to each term inside the polynomial
= x^2 (6) + 4x (6) - 3 (6)
= 6x^2 + 24x - 18

Answer 40

Why wouldn't this one be 4x^2 - 3 * 3x^2 + 6.

Answer 41

Write the whole problem like this..(x^2 + 4x – 3)(2x^2 + x + 6)

Answer 42

Then go from there showing me each step I have to do..

Answer 43

(x^2 + 4x - 3)(2x^2 + x + 6)
Multiply the (x^2 + 4x - 3) to each term.

= (x^2 + 4x - 3)(2x^2) + (x^2 + 4x - 3)(x) + (x^2 + 4x - 3)(6)
Distribute in each multiplication individually:
* 2x^2 into (x^2 + 4x - 3)
* x into (x^2 + 4x - 3)
* 6 into (x^2 + 4x - 3)

= (x^2 (2x^2) + 4x(2x^2) + (-3)(2x^2)) + (x^2(x) + 4x(x) + (-3)(x)) + (x^2(6) + 4x(6) + (-3)(6))
Simplify the products by multiplying

= 2x^4 + 8x^3 - 6x^2 + x^3 + 4x^2 - 3x + 6x^2 + 24x - 18
Combine like-terms (I'll reorder it to get the like-terms together first.)

= 2x^4 + (8x^3 + x^3) + (-6x^2 + 4x^2 + 6x^2) + (-3x + 24x) - 18
= 2x^4 + 9x^3 + 4x^2 + 21x - 18

Answer 44

Many of these do sometimes take quite a bit of work to actually push through. This last one is a good example.

Answer 45

I see.

Answer 46

So, this simplified would be all of that?

Answer 47

The simplified form would be: 2x^4 + 9x^3 + 4x^2 + 21x - 18

Also, just curious, what class are you doing this in? Or at least, what is the topic of the section you're doing this stuff in?

Answer 48

Multiplication of Polynomials - Algebra 1.

Answer 49

Ah, okay. Makes sense then. Thanks. :)
Do you kind of understand what I'm doing to simplify in these now?

Answer 50

I do, but I don't. Could you do the last one again without the equation thing and without detailing it. Just show me how you would simplify it and get it in standard form please.

Answer 51

My assignment is this. --> http://gyazo.com/88fb14b23677d4f112bb959accd5a8b5

Answer 52

(x^2 + 4x - 3)(2x^2 + x + 6)
= (x^2 + 4x - 3)(2x^2) + (x^2 + 4x - 3)(x) + (x^2 + 4x - 3)(6)
= (x^2 (2x^2) + 4x(2x^2) + (-3)(2x^2)) + (x^2(x) + 4x(x) + (-3)(x)) + (x^2(6) + 4x(6) + (-3)(6))
= 2x^4 + 8x^3 - 6x^2 + x^3 + 4x^2 - 3x + 6x^2 + 24x - 18
= 2x^4 + (8x^3 + x^3) + (-6x^2 + 4x^2 + 6x^2) + (-3x + 24x) - 18
= 2x^4 + 9x^3 + 4x^2 + 21x - 18

idk, I think the raw work itself is harder to follow personally.

Answer 53

Did you click the link?

Answer 54

Yeah, I see it.
Would you want me to [try to] explain how to do the last two? :)
The last two are much more straight-forward than this one.

Answer 55

Sure. :)

Answer 56

You're gone.

Answer 57

Ugh, chrome crashes again. :\

We are writing an expression for area of a rectangle, which is A=bh.
We are given that b=2x-4 and h=x+5 (Given in the figure)

Thus, by substituting the new information, we get:
A = (2x-4)(x+5)

We are asked to then write it in simplified form. We may distribute twice and add the like terms to simplify. I will choose to distribute (2x-4)
A = (2x-4)(x) + (2x-4)(5)
= 2x(x) - 4(x) + 2x(5) - 4(5)
= 2x^2 - 4x + 10x - 20
= 2x^2 + 6x - 20

Answer 58

I'm still here, not sure why it makes me appear gone after chrome crashes.

Answer 59

I see. There you are! :D

Answer 60

Yeah, hello. :P
You got the last post with the explanation?

Answer 61

Yeah. Write a simplified polynomial expression to represent the area of the rectangle below. So, (2x-4)(x) + (2x-4)(5)
= 2x(x) - 4(x) + 2x(5) - 4(5)
= 2x^2 - 4x + 10x - 20
= 2x^2 + 6x - 20
is a simplified expression?

Answer 62

Yeah. It is in descending degrees / the powers of each x are from greatest-to-least., and I cannot simplify anything in it further.

Answer 63

So, (2x-4)(x) + (2x-4)(5)
= 2x(x) - 4(x) + 2x(5) - 4(5)
= 2x^2 - 4x + 10x - 20
= 2x^2 + 6x - 20
would be my answer? for num. 4?

Answer 64

Standard form means from greatest to least exponents right?

Answer 65

Yeah. (I'd also include the A=(2x-4)(x+5) as the first line, though)

Answer 66

ALright. Now Num. 5..

Answer 67

x-3 .. that's all it gives me..

Answer 68

Okay, #5
We have a square tile. The actual important shape is just this:


Area of a square is A = bh, or A=s^2 (since all sides are equal, b and h are the same).
So, since s=(x+3), we can then write it as
A = (x+3)^2
Of course, the exponent just means to multiply it to itself.
A = (x+3)(x+3)

Hmm.. so, you want to try this one? To try out distributive property here. :)