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A polynomial p(x) = x⁴-2x³+3x²-ax+b, when divided by x-1 gives remainder as 5, when divided by x+1 gives remainder 19. Will this polynomial p(x) be multiple ofx-2? Support your answer with suitable working.

Answer» <html><body><p> </p><h3>answer: hope it works</h3><p> </p><h2>explanation : When f(x) is <a href="https://interviewquestions.tuteehub.com/tag/divided-441039" style="font-weight:bold;" target="_blank" title="Click to know more about DIVIDED">DIVIDED</a> by x-1 and x+1 the remainder are 5 and 19 respectively.</h2><h2>∴f(1)=5 and f(−1)=19</h2><h2>⇒(1) </h2><h2>4</h2><h2> −<a href="https://interviewquestions.tuteehub.com/tag/2-283658" style="font-weight:bold;" target="_blank" title="Click to know more about 2">2</a>×(1) </h2><h2>3</h2><h2> +3×(1) </h2><h2>2</h2><h2> −a×1+b=5</h2><h2>and (−1) </h2><h2>4</h2><h2> −2×(−1) </h2><h2>3</h2><h2> +3×(−1) </h2><h2>2</h2><h2> −a×(−1)+b=19</h2><h2>⇒1−2+3−a+b=5</h2><h2>and 1+2+3+a+b=19</h2><h2>⇒2−a+b=5 and 6+a+b=19</h2><h2>⇒−a+b=3 and a+b=13</h2><h2>Adding these two <a href="https://interviewquestions.tuteehub.com/tag/equations-452536" style="font-weight:bold;" target="_blank" title="Click to know more about EQUATIONS">EQUATIONS</a>, we get</h2><h2>(−a+b)+(a+b)=3+13</h2><h2>⇒2b=16⇒b=8</h2><h2>Putting b=8 and −a+b=3, we get</h2><h2>−a+8=3⇒a=−5⇒a=5</h2><h2>Putting the values of a and b in</h2><h2>f(x)=x </h2><h2>4</h2><h2> −2x </h2><h2>3</h2><h2> +3x </h2><h2>2</h2><h2> −5x+8</h2><h2>The remainder when f(x) is divided by (x-2) is equal to f(2).</h2><h2>So, Remainder =f(2)=(2) </h2><h2>4</h2><h2> −2×(2) </h2><h2>3</h2><h2> +3×(2) </h2><h2>2</h2><h2> −5×2+8=16−16+12−10+8=10</h2></body></html>


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