Question

Find the leading coefficient and the degree of the following polynomial:
p(x)=-3(x-1)^4(3x+1)(5-2x)^3
The lelading term would be (x-1) or would it be 3(x-1)? then wouldn't the leading constant be -3? or 1?

Answer 1

then wouldn't my degree be 4?

Answer 2

hmm I think you have to expand the whole thing first

Answer 3

expand?

Answer 4

\[p(x) = -3(x-1)^{4} (3x + 1) (5 - 2x)^{3}\]
Is this how the expression appears?

Answer 5

yep

Answer 6

well... just the higher exponent binomial, you may need to expand only

Answer 7

Look at the leading term in each sub-expression and just multiply them. That would suffice.

Answer 8

though using the binomial theorem, is rather visible what his the coefficient

Answer 9

(-3) * (x^4) * (3x) * (-2x)^3 should be sufficient to get the leading term of the product.

Answer 10

I do that for every polynomial to get the leading term? How would I then get the degree?

Answer 11

notice what ranga is doing, after expanding each binomial, he's just picking the highest exponent term and multiplying, which will yield the highest exponent from their product

Answer 12

in short, you had to expand all 3 binomials, then multiply them, and grab the highest term for the degree of the polynomial

Answer 13

but a short way to do it, is like ranga did, noticing the binomial theorem, just pick the highest term from each binomial, and multiply them

Answer 14

(-3) * (x^4) * (3x) * (-2x)^3 = (-3)(3)(-2)^3 * (x^4)(x)(x)^3
= (-3)(3)(-8) * (x^4)(x)(x^3)
= 72 * x^8
is the leading term in the product.

Answer 15

okay thank you I am going to look over this for a min

Answer 16

whats the binomial therom?

Answer 17

Sure. The long way to do it will be to multiply the whole thing out using the binomial theorem to expand the fourth power and the third power, then gather all the like terms together and rearrange the polynomial from the highest degree to the lowest. But that will be too consuming. Since all they are interested in is just the leading term you can use the above shortcut.

Binomial theorem lets you find: (a + b)^n.
If n = 2, it is easy: a^2 + 2ab + b^2

But as n increases, such as n = 3, 4, 5, 6, etc. it is hard to remember all those formula. Instead one can remember the binomial theorem and do it for any n.

Answer 18

is that attachment a different leading coefficient?

Answer 19

No.\[\Large a _{n}x^{n}\]is the leading term and \[\Large a_{n}\]is the leading coefficient. But in the attachment the polynomial has been nicely arranged from the highest degree to the lowest degree and so the leading term is easily found.

But here we will have to multiply out everything and gather the like terms,simplify, rearrange from the highest to the lowest degree and only then we will see what the polynomial is. So we have to do this multiplication:
-3 * (x-1) * (x-1) * (x-1) * (x-1) * (3x+1) * (5-2x) * (5-2x) * (5-2x).
But that is too time consuming. We can use binomial theorem but still it will be too time consuming. So I showed you a shortcut if all they want is just the leading term.

Answer 20

ohhh okay. Thank you im going to learn it right now.

Answer 21

what happens to the 1 in (3x+1) and the 5 in (5-2x)^3?

Answer 22

If you really multiply it all out, this is what the polynomial will look like:

72x^8 - 804x^7 + 3666x^6 - 8739x^5 + 11481x^4 - 7746x^3 + 1620x^2 + 825x - 375

As you can see the leading coefficient is 72x^8.

Answer 23

oh my goodness .....

Answer 24

So instead of doing the multiplication we looked at each sub-expression of -3(x-1)^4(3x+1)(5-2x)^3:
(-3) * (x-1)^4 * (3x+1) * (5-2x)^3
The constant (-3) will remain as it is.
(x-1)^4 will have x^4 as the leading term
(3x+1) will have 3x as the leading term
(5-2x)^3 will have (-2x)^3 as the leading term

So if we multiply all of the leading terms of each sub-expression we will know what the leading term of the entire polynomial will be.

Answer 25

how did you get -3?

Answer 26

-3 is in the problem.It is the first thing after the = sign:
p(x)=-3(x-1)^4(3x+1)(5-2x)^3

Answer 27

why would (-2x)^3 be the leading term and not 5?

Answer 28

The phrase "leading term" may be misleading. They use that phrase because normally polynomials are supposed to be arranged from the highest degree to the lowest degree. If that is done, then the phrase "leading term" will be synonymous with "the first term". But sometimes polynomials are not arranged properly and so the leading term will not be the first term but we have to find what is the term that has the highest degree.

In (5-2x)^3 inside the parenthesis the polynomial is not arranged properly. It should have been (-2x + 5)^3 and then you can easily see that -2x is the leading term.

The leading term is always the term that has the highest degree.

Answer 29

sorry one more questions how does -2x have the highest degree?

Answer 30

the polynomial -2x + 5 has two terms. The constant is like degree zero. x^1 is degree 1. So -2x is the leading term.

Answer 31

ohh okay thank you = )

Answer 32

sure. hope things are a bit clearer now.

If you want another problem for practice:
Find the leading term of: 3 * (1 + 2x)^2 * (2 - 5x)

Answer 33

okay brb

Answer 34

ok

Answer 35

the leading term would be 3,2x,and 5x ?

Answer 36

3,2x,and -5x (don't forget the sign)

Answer 37

yes -5x. =)

Answer 38

And you have to multiply all the leading terms remembering the middle term is squared.

Answer 39

2x^2?

Answer 40

The middle term will be (2x)^2

Answer 41

The leading term of 3 * (1 + 2x)^2 * (2 - 5x) will be: 3 * (2x)^2 * (-5x)

(all you have to is just drop everything but the leading term in each sub-expression.)

Answer 42

okay and I take that multiply it together and get the leading coefficient

Answer 43

yes.

Answer 44

nice thank you

Answer 45

you are welcome.