the polynomial 2x4+mx³-6x²+nx+12 leaves a remainder of 9 when divided by(x+3) and a remainder of 44 when divided by(x-2),Find the values of m and n
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the polynomial 2x4+mx³-6x²+nx+12 leaves a remainder of 9 when divided by(x+3) and a remainder of 44 when divided by(x-2),Find the values of m and n
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Answer:
m = 5
n = -8
Step-by-step explanation:
P(x) = 2x⁴ + mx³ - 6x² + nx + 12
d(x) = x + 3
x + 3 = 0
x = -3
P(-3) = 9
P(x) = 2x⁴ + mx³ - 6x² + nx + 12
P(-3) = 2(-3)⁴ + m(-3)³ - 6(-3)² - 3n + 12
9 = 162 - 27m - 54 - 3n + 12
27m + 3n = 111
9m + n = 37 (equation 1)
P(x) = 2x⁴ + mx³ - 6x² + nx + 12
d(x) = x - 2
x - 2 = 0
x = 2
P(2) = 44
P(x) = 2x⁴ + mx³ - 6x² + nx + 12
P(2) = 2(2)⁴ + m(2)³ - 6(2)² + n(2) + 12
44 = 32 + 8m - 24 + 2n + 12
44 - 32 + 24 - 12 = 8m + 2n
4m + n = 12 (equation 2)
9m + n = 37
- 4m + n = 12
___________
5m = 25
m = 5 (Elimination Method)
4m + n = 12
4(5) + n = 12
20 + n = 12
n = -8