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81. Find the remainder when x³ + 3x² + 3x + 1 is divided by
5 + 2x

5 + 2x
5 +2x = 0 $rightwards double arrow$ 2x = - 5 $rightwards double arrow$ x = $negative 5 over 2$
$therefore$ Remainder

$equals open parentheses negative 5 over 2 close parentheses cubed plus 3 open parentheses negative 5 over 2 close parentheses squared plus 3 open parentheses negative 5 over 2 close parentheses plus 1 equals space minus 125 over 8 plus 75 over 4 minus 15 over 2 plus 1 equals negative 27 over 8$

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82. Find the remainder when x³ - ax² + 6x - a is divided by x — a.

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83. Check whether 7 + 3x is a factor of 3x³ + 7x.

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84. The polynomial p(x) = kx³ + 9x² + 4x - 8 when divided by x + 3 leaves a remainder - 20. Find the value of k.

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85. The polynomial p(x) = x⁴ - 2x³ + 3x² - ax + 3a - 7. When divided by (x + 1) leaves the remainder 19. Find the value of a. Also find the remainder, when p(x) is divided by x + 2.

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86. The polynomials ax³ - 3x² + 4 and 2x³ - 5x + a when divided by (x - 2) leave the remainders p and q respectively. If p - 2q = 4, find the value of a.

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