Polynomials (1 Viewer)

coyazayo · Feb 29, 2016

Hey guys reading this thread. As always thanks in advance!

A) Find the remainder when x^100 is divided by (x-1)(x-2) leave answer in index form

B) Find the remainder when x^100-4x^98+5x+ is divided by x^3-2x^2-x+2

InteGrand · Feb 29, 2016

coyazayo said:
Hey guys reading this thread. As always thanks in advance!

A) Find the remainder when x^100 is divided by (x-1)(x-2) leave answer in index form

B) Find the remainder when x^100-4x^98+5x+ is divided by x^3-2x^2-x+2

$\noindent \text{A) Let} \ P(x) = x^{100}. \ \text{Write} \ P(x)=x^{100} \equiv Q(x) (x-1)(x-2) + ax+b \ \text{(division transformation)}, \text{where ax + b is the remainder and Q is a polynomial of degree 98. Now,} \ P(1) = 1^{100} = 1 \Rightarrow 1 = 0 + a\cdot 1 +b \ \text{(from division transformation), so} \ 1 = a+b \quad \ \text{(1). Similarly,} \ P(2) = 2^{100} \Rightarrow 2a+b = 2^{100} \quad \ \text{(2). Solve (1) and (2) simultaneously to find a and b, and then the remainder is ax + b.}$

$\noindent \text{B) Let} \ D(x) = x^3 -2x^2 -x +2. \ \text{Note that} \ D(1) = 0, \ \text{so} \ (x-1) \ \text{is a factor of D(x). Similarly, we see that} \ D(2)=0=D(-1), \ \text{so} \ D(x) = (x-1)(x-2)(x+1). \ \text{So write} \ P(x) = x^{100} -4x^{98} + 5x +??? \ \text{(I think you left out the constant term by mistake). Then do a similar procedure to above, i.e. let} \ P(x) \equiv Q(x) (x-1)(x-2)(x+1) + ax^2 + bx +c. \ \text{Then sub. in} \ x=-1, 1, 2, \ \text{and obtain three equations in the unknowns a, b, c, which we can then solve.}$

Polynomials (1 Viewer)

coyazayo

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InteGrand

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