Answers

2016-04-19T23:49:51+05:30
Substitute value of x as 2.... Mark it equation 1
substitute value of x as-1/2...Mark it as 2
solving by Equating. . We get value of q..
Make equation 1 equal to zero and put value of q...You will get r,
(you need to use factor theorem)
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2016-04-20T03:25:54+05:30
If (A) and (B) there are factors, the polynomial is as follows:
f(x) = qx² + px + r = q(A)(B)

For all  x ∈ R:
qx² + 5x + r    ≡   q(x - 2)(x - ¹/₂)
qx² + 5x + r    ≡   q(x² - ¹/₂x - 2x + 1) = qx² - ⁵/₂qx + q
In order polynomials are equal, all relevant factors (before x², x...) must be equal:
before x² are the same: q
before x:   5 = -2.5q       q must be equal to -2
without x:   r = q
... what was to prove.

Incidentally, our polynomial is as follows:
f(x) = -2x² + 5x -2 = -2(x - 2)(x - ¹/₂)
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