23 Jan Lesson Summary

Mr Johari's Interesting question...


(a) Solve the equation 2x^3 - 7x^2 - 7x + 30 = 0

2x^3 - 7x^2 - 7x + 30 = 0

(x + 2)(2x - 5)(x - 3) = 0

x = -2 or 3 or 2.5


(b) When the expression x^2 + bx + c is divided by x-2, the remainder is R. When the expression is divided by x+1, the remainder is also R

(i) Find the value of b

let f(x) = x^2 + bx + c

f(2) = f(-1)

4 + 2b + c = 1 - b + c

b = -1


(ii) When the expression is divided by x-4, the remainder is 2R. Find the value of c and R

f(4) = 2f(2)

16 + 4b + c = 2(2 + 2b + c)

sub b

16 - 4 + c = 2(4 - 2 + c)

c = 8

R = f(2) = (4 - 2 + 8) = 10

(iii) When the equation is divided by x-t, the remainder is 5R. Find the two possible values of t

f(t) = 5 x 10

t^2 + tb + c = 50

t^2 - t + 8 = 50

t = 7 or -6

(c)

the sketch shows part of the graph y = x^3+px^2+qx+r where p, q, and r are constants.
The points A, B, and C have co-ordinates (-2,0), (2,0) and (4,0) respectively.
Find p, q, r

let f(x) = x^3 + px^2 + qx + r

f(-2) = 0

f(2) = 0

f(4) = 0

Therefore: f(x) = (x + 2)(x - 2)(x - 4)
f(x) = x^3 - 4x^2 - 4x + 16

comparing coef.

-4x^2 = px^2
p = -4

-4x = qx
q = -4

r = 16