The product, P(n), of two numbers is represented by the expression 2n2 - 4n + 3, where n is a real number. a) If one of the numbers is represented by n - 3, what expression represents the other number?
b) What are the two numbers if n = 1?

m = -2 and n = 10

Step-by-step explanation:

If the polynomial mx^3 - 3x^2 + nx + 2 is divided by x + 3, the remainder is -1. Then;

x+ 3 = 0

x = -3

Substitute x =-3 into the polynomial and equate to -1

-1 = m(-3)³-3(-3)²+n(-3) + 2

-1 = -27m - 27 - 3n + 2

27m + 3n = -25 + 1

27m + 3n = -24 ... 1

Similarly, if it is divided by x - 2, the remainder is -4, then;

-4 =  m(2)³-3(2)²+n(2) + 2

-4 = 8m - 12- 2n + 2

8m -2n = -4 + 10

8m + 2n = 6 ... 2

Solve 1 and 2 simultaneously

27m + 3n = -24 ... 1 * 2

8m + 2n = 6 ... 2 * 3

54m + 6n = -48

24m + 6n = 12

54m - 24m = -48 - 12

30m = -60

m = -60/30

m = -2

Get n by substituting m = -2 into 1

From 1:  27m + 3n = -24

27(-2) + 3n = -24

-54 + 3n = -24

3n = -24 +54

3n = 30

n = 30/3

n = 10

Hence m = -2 and n = 10

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