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).

enter image source here

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.