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Confused Polynomial

i've seen this kind of problem but i forgot how to solve...someone please help me

p(x) is a polynomial of degree 3 with p(1)=2, p(2)=3, p(3)=4, and p(4)=6. One of the factor of p(x+2) is ...

The answer is in the form of (x+a) with a is integer number

Note by Dina Andini Sri Hardina
3 years, 5 months ago

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If \(p(x)\) is cubic, we can generalize it as \(p(x) = ax^3 + bx^2 + cx + d\). So, we can plug in our four data points to get four equations in four variables. Namely,

\(a + b + c + d = 2\)

\(8a + 4b + 2c + d = 3\)

\(27a + 9b + 3c + d = 4\)

\(64a + 16b + 4c + d = 6\)

We could solve these linear equations a whole host of ways. One of the easiest is to make this into a matrix and get it into reduced row eschelon form. Here's the Wolfram Alpha input for that: http://www.wolframalpha.com/input/?i=rref%28%5B1%2C+1%2C+1%2C+1%2C+2%5D%3B%5B8%2C+4%2C+2%2C+1%2C+3%5D%3B%5B27%2C+9%2C+3%2C+1%2C+4%5D%3B%5B64%2C+16%2C+4%2C+1%2C+6%5D%29

This gives us solutions of \(a = \frac16, b = -1, c = \frac{17}{6}, d = 0\)

Thus, \(6p(x) = x^3 - 6x^2 + 17x\) -- note that we can multiply \(p(x)\) by \(6\) and still have the same roots.

This factors: \(x(x^2 - 6x + 17)\). Thus, one factor is \(x\), and that is indeed the only integer factor. Michael Tong · 3 years, 5 months ago

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@Michael Tong That is the approach that i was trying to outline. But your presentation makes it easy to understand. But 6*p(x) = x^3+5x+18 with x+2 as one of the factors Sundar R · 3 years, 5 months ago

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We have 4 given values of the coefficients and 4 unknowns (the coefficients a, b, c and d). One could then use formal methods such as matrix inversion to get the value of the coefficients (a=1/6, b = -1, c=17/6 and d =0) Then p(x+2) = (x^2-5x+18)/6 which has a factor of (x+2).

Alternatively, one could just subtract the equation for p(1) from that for p(2) to remove d and get an equation in 3 variables.

p(1) = a + b + c + d = 2
p(2) = 8a + 4b + 2c + d = 3 p(3) = 27a + 9b + 3c + d = 4 p(4) = 64a + 16b + 4c + d = 6 (2)-(1) gives 7a + 3b + c = 1 (5) (3)-(2) gives 19a + 5b + c = 1 (6) (6) -(2) gives 12a + 2b = 0 2b = -12a b = -6a From (5), 7a -18a + c = 1 c = 11a+1 a + b + c + d = 2 a -6a + 11a + 1 + d = 2 6a + 1 + d = 2 d = 1-6a 64a + 16b + 4c + d = 6 64a -96a + 44a + 4 + 1-6a = 6 6a+5=6 6a=1 a = 1/6 b = -1 c = 11/6 + 1 1/6x^3 -x^2 + 17/6x x^3-6x^2+17x (x+2)^3 -6(x+2)^2 + 17(x+2) = x^3 + 8 + 6x(x+2) - 6(x^2+4x+4) + 17x + 34 = x^3+8+6x^2+12x - 6x^2-24x - 24 + 17x + 34 x^3 + 5x + 18 Sundar R · 3 years, 5 months ago

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@Sundar R Unfortunately, my solution not been presented / displayed as i would have liked and possibly, it looks confusing Sundar R · 3 years, 5 months ago

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@Sundar R i've tried this way but i already gave up Dina Andini Sri Hardina · 3 years, 5 months ago

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Hi Dina!

For problems like these consider a polynomial \(H(x)=P(x)-(x+1)\), since \(deg(P(x))=3\), so \(deg(H(x))=3\). Plugging in \(x=1,2,3\) in \(H\), gives \(0\) each time. Since nothing is known about the leading co-efficient of \(P\), so let it be a real number \(c\). So, \[H(x)=c(x-1)(x-2)(x-3)\], giving \[P(x)-(x+1)=c(x-1)(x-2)(x-3)\], now plug in \(x=4\) in the above equation and using the fact that \(P(4)=6\), \(c=\frac{1}{6}\). So \[P(x)=\frac{1}{6}(x-1)(x-2)(x-3)+(x+1)\]. So \[P(x+2)=\frac{x^{3}-x+6x+18}{6}\]. You can now find the factor.................I hope this has helped you. Jit Ganguly · 3 years, 5 months ago

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@Jit Ganguly but why do u use h(x)=p(x)-(x+1)? Dina Andini Sri Hardina · 3 years, 5 months ago

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@Dina Andini Sri Hardina I took this polynomial because of the values given. Jit Ganguly · 3 years, 5 months ago

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@Jit Ganguly very very help jit...anyway thanks:D Dina Andini Sri Hardina · 3 years, 5 months ago

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Sry...but what we have to calculate in the question? Aabhas Mathur · 3 years, 5 months ago

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@Aabhas Mathur One of the factor of p(x+2) in the form x+a Dina Andini Sri Hardina · 3 years, 5 months ago

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@Dina Andini Sri Hardina Jit has done it beautifully. I can also say that u should try to observe a pattern in the values given and then try to define a function. Aabhas Mathur · 3 years, 5 months ago

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@Aabhas Mathur Thanks for the compliment mate, yeah observing the pattern is the key to such problems. Jit Ganguly · 3 years, 5 months ago

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One of the factors is x+2 Sundar R · 3 years, 5 months ago

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