Show that:
(i) $x - 2 i s a f a c t o r o f 5 x^{2} + 15 x - 50$
(ii) $3 x + 2 i s a f a c t o r o f 3 x^{2} - x - 2$

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Solution

(x – a) is a factor of a polynomial f(x) if the remainder, when f(x) is divided by (x – a), is 0, i.e., if f(a) = 0.
$left parenthesis i right parenthesis f left parenthesis x right parenthesis equals 5 x squared plus 15 x minus 50 f left parenthesis 2 right parenthesis equals 5 left parenthesis 2 right parenthesis squared plus 15 left parenthesis 2 right parenthesis minus 50 equals 20 plus 30 minus 50 equals 0 H e n c e space x minus 2 space space i s space a space f a c t o r space o f space 5 x squared plus 15 x minus 50 left parenthesis i i right parenthesis f left parenthesis x right parenthesis equals 3 x squared minus x minus 2 f open parentheses fraction numerator negative 2 over denominator 3 end fraction close parentheses equals 3 open parentheses fraction numerator negative 2 over denominator 3 end fraction close parentheses squared minus open parentheses fraction numerator negative 2 over denominator 3 end fraction close parentheses minus 2 equals 4 over 3 plus 2 over 3 minus 2 equals 2 minus 2 equals 0 H e n c e space 3 x plus 2 space i s space a space f a c t o r space o f space 3 x squared minus x minus 2$

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Using the FActor theroem, show that:
(i) (x-2) is a factor fo $x^{3} - 2 x^{2} - 9 x + 18$ Hence factories the expression
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(iii) (3x+2) is a factor fo $3 x^{3} + 2 x^{2} - 3 x - 2.$ Hence factorise the expression
$3 x^{3} + 2 x^{2} - 3 x - 2$ completely

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(v) $x^{2} + 3 x + 1$ at x = -1/3
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lim x \to 1 \frac{x^{4} - 3 x^{2} + 2}{x^{3} - 5 x^{2} + 3 x + 1}

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Show that: (i) x−2isafactorof5x2+15x−50 (ii) 3x+2isafactorof3x2−x−2

(x – a) is a factor of a polynomial f(x) if the remainder, when f(x) is divided by (x – a), is 0, i.e., if f(a) = 0.

Show that:
(i) $x - 2 i s a f a c t o r o f 5 x^{2} + 15 x - 50$
(ii) $3 x + 2 i s a f a c t o r o f 3 x^{2} - x - 2$