Polynomial Functions?

Question

FInd: p(m+2) if p(x)=3x-8x^2+x^3
This is an example in the book and I don't understand how it was worked out. Could someone explain it for me so I can do my homework problems. I am homeschooled=no teacher.
Solution:
p(m+2)=3(m+2)-8(m+2)^2+(m+2)^3
=3m+6-8m^2-32m-32+m^3+6m^2+12m+8 -- don't understand this part--- 32+m^3+6m^2+12m+8 (last of above solution). How do they come up with this?
Here is the entire Solution the book gives:
p(m+2)=3(m+2)-8(m+2)^2+(m+2)^3 --original
=3m+6-8m^2-32m-32+m^3+6m^2+12m+8 --next
=m^3-2m^2-17m-18 -- answer

It cut off my equations.

Answer 1

Everyone here gave very good detailed answers, but if you need a simplified explanation, here it is:

Basically, wherever you see an "x" in the equation, you will replace it with "(m+2)." You seem to already understand that part...

The part you pointed out was:
=3m+6-8m^2-32m-32+m^3+6m^2+12m

If you write it all out:
p(m+2)=3(m+2)-8(m+2)^2+(m+2)^3

= 3(m+2) - 8(m+2)(m+2) + (m+2)(m+2)(m+2)

= (3m+6) - (8m+16)(m+2) + (m^2+2m+2m+4)(m+2)

= (3m+6) - (8m^2 + 16m + 16m + 32) + (m^3 + 2m^2 + 2m^2 + 4m + 2m^2 + 4m + 4m +8)

= (3m+6) - (8m^2 + 32m + 32) + (m^3 + 6m^2 + 12m +8)

= 3m + 6 - 8m^2 - 32m - 32 + m^3 + 6m^2 + 12m +8

= m^3 - 2m^2 - 17m - 18

Some people can calculate steps in their head and write it out in the short version, but this will look like missing information to you. Write out each and every step until you understand it well, and then you can work out the math in your head.

Answer 2

p(x) = 3x - 8x² + x³.

To find p(m + 2), substitute (m + 2) in for (x) in the function.
p(m + 2) = 3(m + 2) - 8(m + 2)² + (m + 2)³.

Your book solution, I think, would be more helpful if they didn't skip the next step. You know exponents are evaluated before multiplications, so expand the square and the cube first, before distributing.
p(m + 2) = 3(m + 2) - 8(m + 2)² + (m + 2)³.
p(m + 2) = 3(m + 2) - 8(m² + 4m + 4) + (m³ + 6m² + 12m + 8).
p(m + 2) = 3m + 6 - 8m² - 32m - 32 + m³ + 6m² + 12m + 8.

Now combine like terms and you're done.
p(m + 2) = 3m + 6 - 8m² - 32m - 32 + m³ + 6m² + 12m + 8.
p(m + 2) = + m³ - 2m² -17m - 18.

Answer 3

p(x) = 3x - 8x^2 + x^3

x=m+2

let's substitute term per term:

x = m+2
x^2 = (m+2)^2
= (m+2) (m+2)
= m^2 + 4m +4
x^3 = x^2 * x
= (m^2 + 4m +4) (m+2)
= m(m^2 + 4m +4) + 2 (m^2 + 4m +4)
= (m^3 + 4m^2 +4m) +(2m^2 + 8m +8)
= m^3 + 6m^2 +12m +8

now, substitute back in the original equation:

p(m+2) = 3(m+2) - 8(m^2 + 4m + 4) + (m^3 + 6m^2 +12m + 8)
= 3m + 6 - 8m^2 - 32m - 32 + m^3 + 6m^2 +12m + 8
= m^3 - 2m^2 -17m -18

Good to know you are learning this on your own... good luck, and I hope you understand this. :)

Answer 4

p(m + 2) = 3(m + 2) - 8(m + 2)^2 + (m + 2)^3
p(m + 2) = (m + 2)(3 - 8(m + 2) + (m + 2)^2)
p(m + 2) = (m + 2)(3 - 8m - 16 + ((m + 2)(m + 2)))
p(m + 2) = (m + 2)(-13 - 8m + (m^2 + 4m + 4))
p(m + 2) = (m + 2)(-13 - 8m + m^2 + 4m + 4)
p(m + 2) = (m + 2)(m^2 - 4m - 9)
p(m + 2) = m^3 - 4m^2 - 9m + 2m^2 - 8m - 18
p(m + 2) = m^3 - 2m^2 - 17m - 18

-------------------------------------------------------------------

I will do it the long way

p(m + 2) = 3(m + 2) - 8(m + 2)^2 + (m + 2)^3
p(m + 2) = 3m + 6 - 8((m + 2)(m + 2)) + ((m + 2)(m + 2)(m + 2))
p(m + 2) = 3m + 6 - 8(m^2 + 4m + 4) + ((m + 2)(m^2 + 4m + 4))
p(m + 2) = 3m + 6 - 8m^2 - 32m - 32 + (m^3 + 6m^2 + 12m + 8)
p(m + 2) = -29m - 26 - 8m^2 + m^3 + 6m^2 + 12m + 8
p(m + 2) = m^3 - 2m^2 - 17m - 18

Answer 5

you have to get your mind around the idea of a function

this problem is about the function p(x)

that means there is a mathmatical thing you do to any number or expression that falls into the p function

its like a box called p of x, where p is the box and x is the thing you put in the front end, the box is the math expression being worked on the thing, and the result comes out the other end of the box

in this case p of x is writtne p(x)=3x-8x^2+x^3

that means whatever you choose for "x" in the parenthesis after the p, needs to put in place of every "x" in the polynomial expression and evaluated

so, for p(m+2), everywhere there is an "x" in the expression, you have to replace it with (m+2)

that gives you your first equation

p(m+2)=3(m+2)-8(m+2)^2+(m+2)^3

notice that every place there was an "x", now there is an (m+2)

now its just regular algebra

if you don't know how to expand a term like (m+2)^3 you may need to hit the book for awhile and get that down

(m+2)^3=(m+2)*(m+2)*(m+2)

you have to carefully use the distributive property

in the evaluation of the whole function, be careful with your order of operations, evalute the exponants before multiplying the coeficients

follow along the answer you provided above and you will see the algebra flow down to the final answer

if you cannot follow any of the algebra, you are jumping ahead, go back a few dozen (or a few hundred depending on your book) pages and practice solving simple algebraic equations and move through till you can follow the above

good luck
math is cool
math is worth it

Answer 6

p(m+2)=3(m+2)-8(m+2)^2+(m+2)^3 [you have plugged in (m+2) for x]
Now expand (m+2)^2,the second term as m^2+2*m*2+2^2
using the identity (a+b)^2=a^2+2*ab+b^2
and expand (m+2)^3 using the identity
(a+b)^3=a^3+3a^2b+3ab^2+b^3
m^3+3*m^2*2+3*m*2^2+2^3
now remove the brackets
3(m+2)=3m+6...................................................1st term
-8(m+2)^2=-8(m^2+4m+4)=-8m^2-32m-32......2nd term
(m+2)^3=m^3+6m^2+12m+8............................3rdterm
now the expression is
3m+6-8m^2-32m-32+m^3+6m^2+12m+8
now group all the like terms in the descending order of the power
m^3-8m^2+6m^2+3m-32m+12m+6-32+8
m^3-2m^2-17m-18 and bingo you have the answer
now compare it with what your book gives and do let me know whether you understood

Here is the entire Solution the book gives:
p(m+2)=3(m+2)-8(m+2)^2+(m+2)^3 --original
=3m+6-8m^2-32m-32+m^3+6m^2+12m... --next
=m^3-2m^2-17m-18 -- answer

Answer 7

the degree of a polynomial is the optimal exponent of any variable interior the polynomial. the 1st one has a term it quite is -2x^2*x*x it quite is -2x^4 the exponent is 4; so is the degree. 4(x+3)^3 has an x^3 term, ergo degree 3 i dont comprehend if that helped, yet frequently for determining the degree, the 1st subject to do is to multiply your polynomial out. that it quite is multiply the standards together till you have not any parentheses. then order the words in descending order, meaning the 1st term is ax^n, the 2d is bx^(n-a million), and the final term is a few consistent c now it quite is user-friendly: the degree is n. by using the way, having accomplished all of that, you're additionally waiting to do a gaggle of alternative functional polynomial issues.

Answer 8

Basically, just replace X with M+2

An easier example to visualize it with would be:

f(x) = x^2+1

Now f(m+2) = (m+2)^2+1

Basically what happens is if you plotted f(x), you get a parabola shifted up, but if you plot f(m+2), you would get a parabola shifted up by one and to the right by two as well.

But if you don't want to visualize it and just want to know how to do the math, it's just the basic example, except with more terms to evaluate.