Question

Example 12: Find the value of k if (x - 2) is a
factor of x3 + 2x2 - kx + 10. Hence, determine
whether (x + 5) is also a factor.​

Answer 1

$Let \: p(x) = x^{3}+2x^{2}-kx+10$

$i) Given \:(x-2) \:is \:a \:factor \:of \: p(x) \: then$

$p(2) = 0$

$\implies 2^{3} + 2\times 2^{2} - k \times 2 + 10 = 0$

$\implies 8 + 8 - 2k + 10 = 0$

$\implies 26 - 2k = 0$

$\implies - 2k = -26$

$\implies k = \frac{-26}{-2}$

$\implies k = 13$

$\therefore \red { Value \:of \: k } \green {= 13}$

$ii ) Find \: p(-5) = (-5)^{3} + 2\times (-5)^{2} - 13 \times (-5) + 10$

$= -125 + 50 + 65 + 10$

$= -125 + 125$

$= 0$

$p(-5) = 0$

$\therefore\green { (x+5) \:is \: a \: factor \:of \:p(x)}$

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