document.write( "Question 46740: Expand using the binomial theorem: (2x^2-y)^4 \n" ); document.write( "
Algebra.Com's Answer #30944 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
Expand using the binomial theorem: (2x2-y)4\r
\n" );
document.write( "\n" );
document.write( "To expand (A + B)n by the binomial theorem\r
\n" );
document.write( "\n" );
document.write( "Start out with the 1st term, which is 1AnB0\r
\n" );
document.write( "\n" );
document.write( "then:\r
\n" );
document.write( "\n" );
document.write( "1. Multiply the numerical coefficient by the exponent of A,
\n" );
document.write( "2. Divide that by the number of term
\n" );
document.write( "3. Write that number down to start the next term
\n" );
document.write( "4. Beside that write A with an exponent that is 1 less
\n" );
document.write( "5. Beside that write B with an exponent that is 1 more.\r
\n" );
document.write( "\n" );
document.write( "This will be the next term.  Then repeat the process with
\n" );
document.write( "the new term. You are finished when the
\n" );
document.write( "exponent of A becomes 0 and the exponent of B becomes n\r
\n" );
document.write( "\n" );
document.write( "To expand (2x2 - y)4\r
\n" );
document.write( "\n" );
document.write( "Start out with the 1st term followed by a + sign.\r
\n" );
document.write( "\n" );
document.write( "So we begin with 
\n" );
document.write( " 
\n" );
document.write( "1(2x2)4(-y)0 +\r
\n" );
document.write( "\n" );
document.write( "1. Multiply the numerical coefficient 1 by the exponent of (2x2),
\n" );
document.write( "   which is 4.  That gives 1×4 or 4
\n" );
document.write( "2. Divide that by the number of term
\n" );
document.write( "   The number of term is 1, since it's the 1st term, so we divide
\n" );
document.write( "   4 by 1 and get 4\r
\n" );
document.write( "\n" );
document.write( "3. Write that number down to start the next term
\n" );
document.write( "   So we have\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4\r
\n" );
document.write( "\n" );
document.write( "4. Beside that write (2x2) with an exponent that is 1 less than 4
\n" );
document.write( "   So we now have 
\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3\r
\n" );
document.write( "\n" );
document.write( "5. Beside that write (-y) with an exponent that is 1 more than 0
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1\r
\n" );
document.write( "\n" );
document.write( "The exponent of (2x2) has not reached 0 so we put a + sign
\n" );
document.write( "repeat the process\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 +\r
\n" );
document.write( "\n" );
document.write( "No we repeat the process:\r
\n" );
document.write( "\n" );
document.write( "1. Multiply the numerical coefficient 4 by the exponent of (2x2),
\n" );
document.write( "   which is 3.  That gives 4×3 or 12
\n" );
document.write( "2. Divide that by the number of term
\n" );
document.write( "   The number of term is 2, since it's the 2nd term, so we divide
\n" );
document.write( "   12 by 2 and get 6\r
\n" );
document.write( "\n" );
document.write( "3. Write that number down to start the next term
\n" );
document.write( "   So we have\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6\r
\n" );
document.write( "\n" );
document.write( "4. Beside that write (2x2) with an exponent that is 1 less than 3
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2\r
\n" );
document.write( "\n" );
document.write( "5. Beside that write (-y) with an exponent that is 1 more than 1
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 \r
\n" );
document.write( "\n" );
document.write( "The exponent of (2x2) has not reached 0 so we put a + sign
\n" );
document.write( "repeat the process\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 +\r
\n" );
document.write( "\n" );
document.write( "Now we repeat the process:\r
\n" );
document.write( "\n" );
document.write( "1. Multiply the numerical coefficient 6 by the exponent of (2x2),
\n" );
document.write( "   which is 2.  That gives 6×2 or 12
\n" );
document.write( "2. Divide that by the number of term
\n" );
document.write( "   The number of term is 3, since it's the 3rd term, so we divide
\n" );
document.write( "   12 by 3 and get 4\r
\n" );
document.write( "\n" );
document.write( "3. Write that number down to start the next term
\n" );
document.write( "   So we have\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4\r
\n" );
document.write( "\n" );
document.write( "4. Beside that write (2x2) with an exponent that is 1 less than 2
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1\r
\n" );
document.write( "\n" );
document.write( "5. Beside that write (-y) with an exponent that is 1 more than 2
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1(-y)3\r
\n" );
document.write( "\n" );
document.write( "The exponent of (2x2) has not reached 0 so we put a + sign
\n" );
document.write( "repeat the process\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1(-y)3 +\r
\n" );
document.write( "\n" );
document.write( "Now we repeat the process:\r
\n" );
document.write( "\n" );
document.write( "1. Multiply the numerical coefficient 4 by the exponent of (2x2),
\n" );
document.write( "   which is 1.  That gives 4×1 or 4
\n" );
document.write( "2. Divide that by the number of term
\n" );
document.write( "   The number of term is 4, since it's the 4th term, so we divide
\n" );
document.write( "   4 by 4 and get 1\r
\n" );
document.write( "\n" );
document.write( "3. Write that number down to start the next term
\n" );
document.write( "   So we have\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1(-y)3 + 1\r
\n" );
document.write( "\n" );
document.write( "4. Beside that write (2x2) with an exponent that is 1 less than 1
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1(-y)3 + 1(2x2)0\r
\n" );
document.write( "\n" );
document.write( "5. Beside that write (-y) with an exponent that is 1 more than 3
\n" );
document.write( "   So we now have \r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1(-y)3 + 1(2x2)0(-y)6\r
\n" );
document.write( "\n" );
document.write( "The exponent of (2x2) has now reached 0 so we are done except for
\n" );
document.write( "simplifying.  Remember that a number raised to the 0 power is 1.\r
\n" );
document.write( "\n" );
document.write( "1(2x2)4(-y)0 + 4(2x2)3(-y)1 + 6(2x2)2(-y)2 + 4(2x2)1(-y)3 + 1(2x2)0(-y)4\r
\n" );
document.write( "\n" );
document.write( "1(24x8)(1) + 4(23x6)(-y) + 6(22x4)y2 + 4(2x2)(-y3) + 1(1)y4\r
\n" );
document.write( "\n" );
document.write( "16x8 - 32x6y + 24x4y2 - 8x2y3 + y4 \r
\n" );
document.write( "\n" );
document.write( "Edwin \n" );
document.write( "
\n" );