To have roots as described, that means we have the following factors: From multiplicity 2 at x=1 has (x-1)^2 as its factor From multiplicity 1 at x=0 has x as a factor From multiplicity 1 at x = -4 has a factor of x+4 Putting these together we get that P(x) = A (x) (x+4) (x-1)^2 Multiply these out and find P(x) = A (x^2 + 4x) (x^2 - 2x + 1) A ( x^4 - 2x^3 + x^2 + 4x^3 - 8x^2 + 4x ) Combine like terms and find P(x) = A (x^4 + 2x^3 - 7x^2 + 4x) To find A, we use the point they gave us (5, 72) P(5) = A [ (5)^4 + 2(5)^3 - 7(5)^2 + 4(5) ] = 72 A [ 625 + 250 - 175 + 20 ] = 72 A [ 720 ] = 72 Divide both sides by 720 and find that A = 0.1 Final answer: P(x) = 0.1 ( x^4 + 2x^3 - 7x^2 + 4x) or P(x) = 0.1 x^4 + 0.2 x^3 - 0.7x^2 + 0.4x
Polynomial which passes through (5,9) is P(x) = 0.3x⁴ - 2.1x³+ 2.4x² +4.8x
"Polynomial is an algebraic equation which express variables with different exponents and coefficient."
According to the question,
P(x) has a root of multiplicity 2 at x=4 is express as
(x-4)²
P(x) has roots of multiplicity 1 at x=0 and x=-1 is express as
(x-0) and (x+1)
Combine all the given conditions we get,
P(x) = A[ (x-4)²(x-0)(x+1)
= A [ (x² -8x +16)(x²+x)]
= A [x⁴ - 7x³+8x² +16x]
As per the given condition it passes through (5,9)
Substitute the value in above equation we get,
9 = A [ 5⁴ - 7(5)³+8(5)²+16(5)]
⇒ 9 = A(30)
⇒A = 3 /10
⇒A = 0.3
Therefore,
P(x) = 0.3 [ x⁴ - 7x³+8x² +16x]
= 0.3x⁴ -2.1x³+2.4x² +4.8x
Hence, polynomial which passes through (5,9) is P(x) = 0.3x⁴ - 2.1x³+ 2.4x² +4.8x
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