Question

What must be subtracted from 14x3- 5x2 + 9x + 7 So that the resulting polynomial is exactly divisible by 2x – 1.

Answer 1

Solution :

Division :

$\boxed{\begin{array}{l|n|r}\sf 2x-1 & \sf 14x^{3} -5x^{2}+9x+7& \sf 7x^{2}+x+5\\ & \sf 14x^{3}-7x^{2} \\ & \:\:(-)\:\:(+)\\ &\rule{90}{0.8}\\&\sf \qquad\quad 2x^{2} + 9x+7\\ & \sf \qquad\quad 2x^{2} - x\\ &\qquad\quad(-)\:\:(+) \\ &\qquad\quad\rule{70}{0.8}\\ &\sf \qquad \qquad\quad 10x + 7 \\ & \sf \qquad \qquad \quad10x-5 \\ &\qquad\qqaud\quad\:\:\:\:\:\:(-)\:\:(+) \\ &\qquad\qquad \rule{60}{0.8} \\ & \sf \qquad \qquad\quad \:\:\:\:\:\:\:\:\:\:12 \end{array}}$

∴ 12 is the remainder & also must be subtracted from 14x³ - 5x² + 9x + 7 that the resulting polynomial is exactly divisible by 2x - 1.