document.write( "Question 1209336: Let $b$ be a constant. What is the smallest possible degree of the polynomial $f(x) + b\cdot g(x)$? Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1. \n" ); document.write( "

Algebra.Com's Answer #848511 by ikleyn(52786) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Let $b$ be a constant. What is the smallest possible degree of the polynomial f(x) + b*g(x)?
\n" ); document.write( "Let f(x) = x^4-3x^2 + 2 and g(x) = 2x^4 - 6x^2 + 2x -1.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Regularly, we expect that f(x) + b*g(x)  is the 4-degree polynomial.   But if you take b= -1/2,\r\n" );
document.write( "\r\n" );
document.write( "the terms containing x^4 will cancel each other; the terms containing x^2 also will cancel each other, \r\n" );
document.write( "\r\n" );
document.write( "and then you will get the sum f(x) + b*g(x) as the polynomial of the degree 1 (one),\r\n" );
document.write( "\r\n" );
document.write( "which is a linear binomial.  So, the smallest possible degree is 1 at b = -1/2.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved, answered and explained.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );