Question

18. Use the factor theorem to determine whether g(x) = x+1 is a factor of p(x) = 2x3 +x2 -2x -1 and verify
p(x)= g(x).q(x)+r(x)
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Answer 1

Solution :

Given :

$\\ \large \sf \: g(x) = x + 1 \\ \\ \large \sf \: x = - 1$

$\large \sf p(x) = 2 {x}^{3} + {x}^{2} - 2x - 1 \\ \\ \large \sf \: \underline{remainder \: \: theorem : } \\ \\ \large \sf : \implies \: p( - 1) \\ \\ \large \tt= 2( { - 1)}^{ 3} + ( - 1 {)}^{2} - 2( - 1) - 1 \\ \\ \large \tt = - 2 + 1 + 2 - 1 \\ \\ \large \tt = - 3 + 3 \\ \\ \large \tt \boxed{ = 0}$

According to pic 1

$\\ \large \sf \: q(x) = 2 {x}^{2} + 3x + 1$

• Splitting The Middle Term

$\\ \\ \large \sf \implies \: 2 {x}^{2} + 3x + 1\\ \\ \large \sf \implies \: 2 {x}^{2} + 2x + x + 1 \\ \\ \large \sf \implies \: 2x(x + 1) + 1(x + 1) \\ \\ \large \sf \mapsto \: r(x) = (2x + 1)(x + 1)$