How do you find the degree, leading term, the leading coefficient, the constant term and the end behavior of #f(x)=sqrt3x^17+22.5x^10-pix^7+1/3#?

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Nimo N. Share
Feb 19, 2018


See below.


How do you find
a) the degree,
b) leading term,
c) the leading coefficient,
d) the constant term and
e) the end behavior of
# color(blue)( f(x) = sqrt(3) x^17 + 22.5x^10 − πx^7 + 1/3 #

a) The degree of a polynomial is the degree of the term with the highest power of the variable.
The highest power present in the polynomial is 17, so the degree of the relation, is # color(brown)( 17 #.

b) The leading term of a polynomial, in standard form, i.e. written with descending powers of the variable, is the term with the highest degree, so the leading term is # color(brown)( √3 x^17 #.

c) The leading coefficient is the coefficient of the leading term, i.e. # color(brown)( sqrt(3) #.

d) The constant term of a polynomial is the term that can be described in one of two ways:
either the term has no variable factor, or
the term has a variable to the zero power as a factor.
The constant term is # color(brown)( 1/3 #.

e) The end behavior is found by looking at what the relation does as the variable goes toward # -oo # and # +oo #, which can be expressed as:
# lim_(x rarr -oo) f(x) #, and # lim_(x rarr +oo) f(x) #,

Examining the given relation, one sees that the leading term overpowers the contributions of the other terms, which contribute little to the value of the relation for large values of x, whether "large and negative" or "large and positive".

To answer this question, then,
# color(brown)( lim_(x rarr -oo) f(x) = -oo #, and
# color(brown)( lim_(x rarr -oo) f(x) = +oo #

To verify the statement made above about the effect of the effect of large values on the terms, calculate each term for # x = 100 #, to see the effect:
# sqrt(3) * (100)^17 ~~ 1.732 * 10^34 #
# + 22.5 (100)^10 = 2.25 * 10^21 #
# − π (100)^7 = - 3.142 * 10^14 #
# + 1/3 ~~ 0.333 #

f(x) has such an interesting graph, I thought it would be good to add it to the display.
enter image source here

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