document.write( "Question 347518: Write the resulting polynomial in standard form. [(x+1)-y]^2 \n" ); document.write( "
Algebra.Com's Answer #248538 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"%28%28x%2B1%29-y%29%5E2\"
\n" ); document.write( "There's a short way and a long way to do this.

\n" ); document.write( "The short way depends on
  • Knowing the pattern: \"%28a-b%29%5E2+=+a%5E2+-+2ab+%2B+b%5E2\" and...
  • Seeing how this pattern can be used on your expression

\n" ); document.write( "The key is understanding that in the pattern for \"%28a+-+b%29%5E2\", the \"a\" and the \"b\" can be any expression. We just need two expressions being subtracted!

\n" ); document.write( "In your expression, you have two expressions being subtracted: (x+1) and y. SO we can use the pattern replacing \"a\" with (x+1) and \"b\" with y:
\n" ); document.write( "\"%28x%2B1%29%5E2+-+2%28x%2B1%29y+%2By%5E2\"
\n" ); document.write( "(Note: If you cannot \"see\" how this was done, then perhaps the use of a temporary variable will help. Let q = (x+1). Substituting q for (x+1) in your expression we get \"%28q+-+y%29%5E2\". It should be obvious how this fits the pattern for \"%28a+-+b%29%5E2\": \"q%5E2+-+2qy+%2B+y%5E2\". Now that we are finished using q to help us see how to use the pattern, we just replace the q with (x+1) giving: \"%28x%2B1%29%5E2+-+2%28x%2B1%29y+%2By%5E2\" (which is the same as the expression we got without using the temporary variable, q.)

\n" ); document.write( "Now we can use the pattern for \"a+%2B+b%29%5E2\" on the first part of the above:
\n" ); document.write( "\"%28x%29%5E2+%2B+2%28x%29%281%29+%2B+1%5E2+-+2%28x%2B1%29y+%2By%5E2\"
\n" ); document.write( "Continuing to simplify:
\n" ); document.write( "\"x%5E2+%2B+2x+%2B+1+-+2y%28x%2B1%29+%2B+y%5E2\"
\n" ); document.write( "\"x%5E2+%2B+2x+%2B+1+-+2xy+-+2y+%2B+y%5E2\"
\n" ); document.write( "Last of all we put it in standard form:
\n" ); document.write( "\"x%5E2+-2xy+%2B+y%5E2+%2B+2x+-+2y+%2B+1\"

\n" ); document.write( "The long way is to actually square your expression:
\n" ); document.write( "\"%28x%2B1-y%29%5E2\"
\n" ); document.write( "\"%28x%2B1-y%29%28x%2B1-y%29\"
\n" ); document.write( "To multiply polynomials we multiply each term of one polynomial by each term of the other polynomial. Since there are 3 terms in each of the polynomials there will be 3*3 or 9 multiplications:
\n" ); document.write( "\"x%2Ax+%2B+x%2A1+-x%2Ay+%2B+1%2Ax+%2B+1%2A1+-+1%2Ay+-+y%2Ax+-+y%2A1+-y%2A%28-y%29\"
\n" ); document.write( "Now we simplify:
\n" ); document.write( "\"x%5E2+%2B+x+-+x%2Ay+%2B+x+%2B+1+-+y+-+xy+-+y+%2B+y%5E2\"
\n" ); document.write( "\"x%5E2+%2B+2x+-+x%2Ay+%2B++1+-+2y+-+2xy++%2By%5E2\"
\n" ); document.write( "And now standard form:
\n" ); document.write( "\"x%5E2+-2xy+%2B+y%5E2+%2B+2x+-+2y+%2B+1\"
\n" ); document.write( "The is the same as we got the short way. \n" ); document.write( "
\n" );