In polynomials (1 Viewer)

[Real Madrid]

For the polynomial P(x)=(a+1)x^3+(b-7)x^2+c+5, find the values for a,b or c if

(a) P(x) is monic
(b) the coefficient of x^2 is 3
(c) the constant term is -1
(d) P(x) has a degree 2
(e) The leading term has a coefficient of 5

Show working

 

[tommykins]

回复: In polynomials

a) a+1 = 1, a = 0.
b) b - 7 = 2, b = 9
c) c+5 = -1, c = -6
d) a+1 = 0, a = -1
e) a+1 = 5, a = 4

 

[Real Madrid]

Can someone explai nthe concpets of leading terms and monics?

 

[Timothy.Siu]

Can someone explai nthe concpets of leading terms and monics?

i'm guessing the leading term is the term with the highest degree
and monic just means the coefficient is 1

 

[tommykins]

回复: Re: In polynomials

Hrm...how to do this in a simple manner.
Let a,b,c,d,e,f..... be real numbers.

A polynomial is defined as P(x) = ax^n + bx^n-1 + cx^n-2 ...+zx^0

From that, we know that any number x^0 is 1. Thus -
P(x) = ax^n + bx^n-1 + cx^n-2 ...+z

Now, a polynomial of degree 'n' means the highest power is x^n.
For example in P(x) = x^3 + 6x^5 + 1, the degree of the polynomial is 6.
The leading term is the numbers a,b,c,d,e....

Monic means the degree of the polynomial has a coefficient of 1.

Let P(x) = ax^n + bx^n-1 + cx^n-2 ...+z
If P(x) was monic, that means a = 1, so P(x) = x^n + bx^n-1 + cx^n-2 ...+z

 

[Real Madrid]

But then what about the values of B and C... can't it just be monic if the coefficient ends up being 1?, so B and C can be 8 and -4 respectively?

 

[tommykins]

回复: Re: In polynomials

Monic only applies to the degree of the polynomial (the leading term).

 

[Real Madrid]

Leading term is the term that has the highest power (degree)?

 

[tommykins]

回复: Re: In polynomials

Yes mate

 

[Real Madrid]

It's always me asking the questions roflmao.

I was overseas awhile ago, and I have to catch up in time to pass my MX1 exam.

 

[Aplus]

It's always me asking the questions roflmao.

I was overseas awhile ago, and I have to catch up in time to pass my MX1 exam.

Yeah, that's basically why my parents have said no overseas trips until you finish your HSC (they said this like last year).