The graph #y=(x^2+5x+7)/(x+2)# is shown below. Find the value of the product #abcd#?

The dashed vertical line through D (d,0) is a vertical asymptote. The dashed line BC is a slant asymptote. The coordinates of B and C are B(0,b) and C(c,0). The curve crosses the y-axis at A(0,a).

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1 Answer
Jul 11, 2018

#abcd=63#

Explanation:

The question asks us to find various properties of the curve and combine them.

1) Vertical asymptote, #d#

Find the vertical asymptote by setting the denominator to 0:
#x+2=0rArrx=-2rArrd=-2#.

Check that the numerator is not also 0 at the same point:
#(-2)^2+5*(-2)+7=4-10+7=1!=0#
So this point is not a function hole.

2) Curve y-intercept, #a#

We seek the function value #y(0)#.

#y(0)=7/2rArra=7/2#.

3) Slant intercept

Use polynomial long division to express #y(x)# as a sum of terms.

Divide leading terms: #x# goes into #x^2# #x# times. So subtract off #x(x+2)=x^2+2x#:
#x^2+5x+7-x^2-2x=3x+7#
Repeat: #x# goes into #3x# #3# times. So subtract off #3(x+2)=3x+6#:
#3x+7-3x-6=1#

So #y(x)=(x^2+5x+7)/(x+2)=x+3+1/(x+2)#, an identity which you can confirm by multiplying back out.

We know from this that the slant asymptote line is #y=x+3# - the fractional term goes to zero at the infinities.

4) Slant intercept #x#- and #y#-intercepts, #c# and #b#

For the #x#-intercept, #y=0#, so
#0=x+3rArrx=-3rArrc=-3#

For the #y#-intercept, #x=0#, so
#y=0+3rArry=3rArrb=3#

5) Combine results

From above, #a=7/2#, #b=3#, #c=-3#, and #d=-2#.

So #abcd=7/2*3*(-3)*(-2)=63#.