Use the Remainder Theorem to find which of the following is a factor of $2 x^{3} + 3 x^{2} - 5 x - 6$
(a) x+1 (ii) 2x-1 (iii) x+2

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Solution

By remainder theorem we know that when a polynomial f (x) is divided by x – a, then the remainder is 0.
Let f(x) = 2x³ + 3x² – 5x – 6
(i) f (-1) = 2(-1)³ + 3(-1)² – 5(-1) – 6 = -2 + 3 + 5 – 6 = 0
Thus, (x + 1) is a factor of the polynomial f(x).
$left parenthesis i i right parenthesis f open parentheses 1 half close parentheses equals 2 open parentheses 1 half close parentheses cubed plus 3 open parentheses 1 half close parentheses cubed minus 5 open parentheses 1 half close parentheses minus 6 equals 1 fourth plus 3 over 4 plus 5 over 6 minus 6 equals negative 5 over 2 minus 5 equals fraction numerator negative 15 over denominator 2 end fraction not equal to 0$

Thus, (2x – 1) is not a factor of the polynomial f(x).
(iii) f (-2) = 2(-2)³ + 3(-2)² – 5(-2) – 6 = -16 + 12 + 10 – 6 = 0
Thus, (x + 2) is a factor of the polynomial f(x).

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Use the Remainder Theorem to find which of the following is a factor of 2x3+3x2−5x−6 (a) x+1 (ii) 2x-1 (iii) x+2

Use the Remainder Theorem to find which of the following is a factor of $2 x^{3} + 3 x^{2} - 5 x - 6$
(a) x+1 (ii) 2x-1 (iii) x+2