If-P-x-is-a-polynomial-whose-sum-of-coefficients-is-3-and-P-x-can-be-factorised-into-two-polynomials-Q-x-R-x-with-integer-coefficients-the-sum-of-the-coefficients-Q-x-2-R-x-2-is-

Question Number 110374 by Aina Samuel Temidayo last updated on 28/Aug/20

If P (x) is a polynomial whose sum of

coefficients is 3 and P (x) can be

factorised into two polynomials

Q (x), R (x) with integer coefficients,

the sum of the coefficients

Q {(x)}^{2} + R {(x)}^{2} is

Answered by mr W last updated on 28/Aug/20

Q (x) = \underset{i = 0}{\sum^{m}} q_{i} x^{i}

R (x) = \underset{j = 0}{\sum^{n}} r_{j} x^{j}

P (x) = Q (x) R (x) = (\underset{i = 0}{\sum^{m}} q_{i} x^{i}) (\underset{j = 0}{\sum^{n}} r_{j} x^{j})

P (1) = (\underset{i = 0}{\sum^{m}} q_{i}) (\underset{j = 0}{\sum^{n}} r_{j}) = 3

\Rightarrow \underset{i = 0}{\sum^{m}} q_{i} = \pm 1, \underset{j = 0}{\sum^{n}} r_{j} = \pm 3 o r \underset{i = 0}{\sum^{m}} q_{i} = \pm 3, \underset{j = 0}{\sum^{n}} r_{j} = \pm 1

s u m o f c o e f . o f Q {(x)}^{2} + R {(x)}^{2} is

Q {(1)}^{2} + R {(1)}^{2} = {(\underset{i = 0}{\sum^{m}} q_{i})}^{2} + {(\underset{j = 0}{\sum^{n}} r_{j})}^{2} = 10

Commented by Aina Samuel Temidayo last updated on 28/Aug/20

Thanks .

Answered by floor(10²Eta[1]) last updated on 28/Aug/20

P (1) = 3

P (x) = Q (x) . R (x)

F (x) = Q {(x)}^{2} + R {(x)}^{2}

F (1) = Q {(1)}^{2} + R {(1)}^{2}

Q (x) = \frac{P (x)}{R (x)} ∴ Q {(x)}^{2} = \frac{P {(x)}^{2}}{R {(x)}^{2}}

\Rightarrow Q {(1)}^{2} = \frac{R {(1)}^{2}}{P {(1)}^{2}} ∴ Q {(1)}^{2} = \frac{R {(1)}^{2}}{9}

F (1) = R {(1)}^{2} . \frac{10}{9}

since Q (x) and R (x) \in Z [x]

\Rightarrow the sum of coefficients of these polynomials

also have to be an integer .

so 3 = R (1) . Q (1)

\Rightarrow R (1) = 3 and Q (1) = 1

or R (1) = 1 and Q (1) = 3

or R (1) = - 1 and Q (1) = - 3

or R (1) = - 3 and Q (1) = - 1

∴ F (1) = \frac{10}{9} R {(1)}^{2} = 10 or \frac{10}{9}

but F (x) \in Z [x] because F (x) = Q {(x)}^{2} + R {(x)}^{2}

so F (1) \in Z \Rightarrow F (1) = 10

Commented by Aina Samuel Temidayo last updated on 28/Aug/20

Thanks .