Question

Determine values of m, n, p, and q so that

Answer 1

Explanation:

`(x^(2)+mx+n )/(9-x^(2) ) /(x^(2) +6x-16)/(x^(2) +px+q) = (-x+3)/(x+8)`

Change the division of the two fractions in to multiplication

`(x^(2)+mx+n )/(9-x^(2) ) * (x^(2) +px+q)/(x^(2) +6x-16) = (-x+3)/(x+8)`

Factor what we can

`(x^(2)+mx+n )/((3-x)(3+x)) * (x^(2) +px+q)/((x-2)(x+8)) = ((3-x))/(x+8)`

Cross multiply and simplify (x+8)/(x+8)

(x² +mx +n) (x² +px +q) = (x -3) (x -3) (x +3)( x -2)

Now we look to see what values for m, n, p, and q will give those factors.

m= 1 n= 2 p= 3 q= 4

(x² +x +2) (x² +3x +4) ≠ (x -3) (x -3) (x +3)( x -2) ❌

m= 1 n= -6 p= -6 q= 9

(x² +x -6) (x² -6x +9) = (x+3)(x-2) (x-3)² ✅

m= -5 n= 6 p= 0 q= -9

(x² -5x +6) (x² -9) = (x-3)(x-2)(x-3)(x+3)✅

m= 7 n= -4 p= 2 q= 0

(x² +7x -4) (x² +2x) ≠ (x-3)(x-2)(x-3)(x+3)❌

m= 3 n= -6 p= 1 q= 7

(x² +3x -6) (x² +x+7) ≠(x-3)(x-2)(x-3)(x+3)❌

m= -6 n= 9 p= 1 q= -6

(x² -6x +9) (x² +x -6) = (x-3)(x-3)(x-2)(x+3)✅

m= 0 n= -9 p= -5 q= 6

(x² -9) (x² -5x +6) ≠(x-3)(x-3)(x+3)(x-2)❌