which syllabus are these questions? are these in secondary school?

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which syllabus are these questions? are these in secondary school?

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The rational factors, if they exist, are one or some of the following :

+- factors of 72 / factors of 4

= +- 1, 2,3,4, 6,8,9,12,18,36,72, +- 1/2, 1/4, 3/2, 3/4, 9/2, 9/4

On checking them you find that

Let us rewrite the polynomial in the following way. and find the value of "a"

(x-3) (x+3/2) 4 (x² + a x - 72/(4*-3*3/2) ) = 0

(x² - 3/2 x - 9/2) 4 (x² + a x + 4) = 0

Coefficient of x³ will be : 4 a - 4 * 3/2 = - 12

a = -3/2

So x² - 3/2 x + 4 = 0

So there are two real and two imaginary roots.

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Trying the rational roots:

P(0) = -ve P(4) = 308 = +ve. so a root exists between x= 0 & 4.

Trying P(3) =0 x = 3 is a root.

P(-4) = 1820 = +ve So a root exists between 0 and -4.

Trying -1, -2, and -3/2 works.