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).

Jul 11, 2018

$a b c d = 63$

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:
$x + 2 = 0 \Rightarrow x = - 2 \Rightarrow d = - 2$.

Check that the numerator is not also 0 at the same point:
${\left(- 2\right)}^{2} + 5 \cdot \left(- 2\right) + 7 = 4 - 10 + 7 = 1 \ne 0$
So this point is not a function hole.

2) Curve y-intercept, $a$

We seek the function value $y \left(0\right)$.

$y \left(0\right) = \frac{7}{2} \Rightarrow a = \frac{7}{2}$.

3) Slant intercept

Use polynomial long division to express $y \left(x\right)$ as a sum of terms.

Divide leading terms: $x$ goes into ${x}^{2}$ $x$ times. So subtract off $x \left(x + 2\right) = {x}^{2} + 2 x$:
${x}^{2} + 5 x + 7 - {x}^{2} - 2 x = 3 x + 7$
Repeat: $x$ goes into $3 x$ $3$ times. So subtract off $3 \left(x + 2\right) = 3 x + 6$:
$3 x + 7 - 3 x - 6 = 1$

So $y \left(x\right) = \frac{{x}^{2} + 5 x + 7}{x + 2} = x + 3 + \frac{1}{x + 2}$, 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
$0 = x + 3 \Rightarrow x = - 3 \Rightarrow c = - 3$

For the $y$-intercept, $x = 0$, so
$y = 0 + 3 \Rightarrow y = 3 \Rightarrow b = 3$

5) Combine results

From above, $a = \frac{7}{2}$, $b = 3$, $c = - 3$, and $d = - 2$.

So $a b c d = \frac{7}{2} \cdot 3 \cdot \left(- 3\right) \cdot \left(- 2\right) = 63$.