Answer to Question #92482 in Calculus for Vaibhavi Desai

Question

Question 1

(i) If (x + 3) is a factor of the
expression x^3 + 3x^2 - x - 3, find the other two factors?

(ii) If the expression above was divided by (x + 2), what is the remainder?

Question 2

(a) Sketch the graphs on the same axis.
(i) y = x^2 - 4
(ii) y = -x + 3

(b) What are the inequalities of the region bounded (enclosed area) by the two graphs.

Answer 1

Question 1

i). x³+3x²-x-3= x²(x+3)-(x+3)=(x+3)(x²-1)=(x+3)(x-1)(x+1)

So, the other two factors are x-1 and x+1.

ii). x³+3x²-x-3= x²(x+2) + x²-x-3=x²(x+2) +x(x+2)-3x-3=(x+2)(x²+x) -3(x+2)+3=(x+2)(x²+x-3)+3

So, the remainder in this case is 3.

Question 2

a) i) and ii) The two functions are drawn on the same system of axis (see attachment).

The two intersection points between the two functions are the solutions of the equations

$x^2-4 = -x+3 => x^2+x-7=0$

b) The inequalities of the area bounded by the two functions are:

$x \ge (-1- \sqrt{29}) /2$

$x \le (-1+ \sqrt{29})/2$

$y \le -x+3$

$y \ge x^2-4$

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