Question

Please anyone answer my question

Show that this function does not have any integer roots :

Q ( x ) = 2 x⁷ + 5 x⁶ - 20 x⁵ - 62 x³ + 15 x² - 60 x + 405

Plz answer fast..........

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Answer 1

Seems interesting o_O

Hey buddy , let's say hello to The Rational Root Theorem

Hey there RRT , what's up ?
Didn't heard ya , can you repeat
Okie ^_^

So , The Rational Root Theorem says that if a polynomial has a rational root , then the numerator of the root must divide the constant term , and the denominator must divide the leading coefficient.

According to the question ,

$Q(x) =2 {x}^{7} + 5 {x}^{6} - 20 {x}^{5} - 62 {x}^{3} + 15 {x}^{2} - 60x + 405$

Each rational solution x can be written as a fraction x = p/q

Where p is an integer factor of the constant term
And q is an integer factor of the leading coefficient

Here ,
Constant term = 405
And leading coefficient = 2

$405 = 1 \times 5 \times {3}^{4}$

Hence ,
q = 2
p = +1 , +3 ,+3² , +3³ , +3⁴ , -1 , -3 ,-3² , -3³ , -3⁴

[ We ain't considering q = 1 here , as that's another case ( a special case of rational root theoram also known as integral root theoram )]

So , let's make the possible cases for x

$x = \frac{p}{q}$

Ah wait ! All possible values of p are odd while that is p is 2 or an even number
Always a fraction ! Can't be reduced to an integer

Hence , Q(x) has no integer roots.

Answer 2

Given

Q(x) = 2 x⁷ + 5 x⁶ - 20 x⁵ - 62 x³ + 15 x² - 60 x + 405

ANSWER

The rational root theorem says that if a polynomial has rational root then the numerator of the root must divide the constant term and the numerator must divide the leading coefficient.

Here in Q(x)=2 x⁷ + 5 x⁶ - 20 x⁵ - 62 x³ + 15 x² - 60 x + 405

Each of the rational solutions can be expressed as x =$\dfrac{p}{q}$.

Here p is an integer factor of the constant term and

q is an integer factor of the leading coefficient.

So, q =2

405 = 1×5 ×$3^4$

=>P=+1,+3,$+3^2$,$3^3$,$3^4$,-1,-3,-3^2,$-3^3$,$-3^4$ and 5.

Note

We are not considering q =1 since it is a special case of root theorem i.e integral root theorem.

X =$\dfrac{p}{q}$.

Hence all possible values of p are odd and q are even.

So,x is never an integer.

Hence PROVED.