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kaliprasad
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Let P(x) be a polynomial of x. Show that there exists a polynomial Q(x) such that P(x)Q(x) is a polynomial of $x^3$
my solution :kaliprasad said:Let P(x) be a polynomial of x. Show that there exists a polynomial Q(x) such that P(x)Q(x) is a polynomial of $x^3$
Albert said:my solution :
else $Q(x)=\dfrac {N(x^3)-R(x)}{P(x)}=M(x^2)$
so this $Q(x)$ always exists
kaliprasad said:in the else part the condition is not met.
Albert said:in this case $Q(x)=\dfrac {N(x^3)-R(x)}{P(x)}$
Albert said:exam :
$x^3+1=(x+1)(x^2-x+1)+R(x), \,\, P(x)=x+1, Q(x)=x^2-x+1,R(x)=0$
$N(x^3)=x^3+1$ is a multiple of $x+1$
$x^3+2=(x+1)(x^2-x+1)+1---(A)$
the coefficient of $(A)\in Z$
$N(x^3)=x^3+2$ is not a multipl of $x+1$
but $x^3+2=x^3+(2^{\frac {1}{3}})^3=(x+2^{\frac {1}{3}})(x^2-2^{\frac {1}{3}}x+2^{\frac {2}{3}})---(B)$
here $P(x)=(x+2^{\frac {1}{3}}),Q(x)=(x^2-2^{\frac {1}{3}}x+2^{\frac {2}{3}})$
the coefficient of $(B)\in R$
the diversity occurs because the coefficients coming from different set
kaliprasad said:if $P(x) = x^2 + 1$ then $ Q(x) = x^4 - x^2 - 1$
if $P(X) = x^2 +x + 1$ then $Q(x) = x - 1$
if $(P(X) = x^2 + x + 3$ then how do you find Q(x)
note: more than one Q(x) may be there and we are interested in anyone
My mistake I meantAlbert said:Do you consider $(x^2+1)(x^4-x^2-1)=x^6-2x^2-1=(x^3)^2-2(x^3)^\frac{2}{3}-1$ as a polynomial of $x^3$?
[sp][This method only works if the coefficients are allowed to be complex numbers.]kaliprasad said:Let P(x) be a polynomial of x. Show that there exists a polynomial Q(x) such that P(x)Q(x) is a polynomial of $x^3$
kaliprasad said:Let P(x) be a polynomial of x. Show that there exists a polynomial Q(x) such that P(x)Q(x) is a polynomial of $x^3$
Polynomial division is a method used to divide one polynomial by another polynomial in order to find the quotient and remainder.
To perform polynomial division, first arrange the polynomials in descending order of degree. Then, divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Multiply this term by the divisor and subtract it from the dividend. Repeat this process until the dividend is completely divided or the degree of the remainder is less than the degree of the divisor.
The quotient in polynomial division is the result of dividing one polynomial by another polynomial. It is the answer to the division problem and is typically written in the form of Q(x).
The remainder in polynomial division is the leftover term after the polynomial division is performed. It is typically written in the form of R(x) and its degree is always less than the degree of the divisor.
Polynomial division is important because it allows us to solve problems involving polynomial functions, such as finding the roots or factors of a polynomial. It is also used in many other areas of mathematics, such as calculus and differential equations.