document.write( "Question 477246: Please help me with those questions. Thanks a lot!!!!!!!!!
\n" ); document.write( "1)Factor x^4 + 4y^4.
\n" ); document.write( "2)What is the coefficient of x in (x^4 + x^3 + x^2 + x + 1)^4?
\n" ); document.write( "3)What is the coefficient of x^3 in (x^4 + x^3 + x^2 + x + 1)^4? \n" ); document.write( "
Algebra.Com's Answer #327179 by Edwin McCravy(20054) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "1)Factor x⁴ + 4y⁴.\r\n" );
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document.write( "That's the sum of two squares.  But we can only factor the\r\n" );
document.write( "difference of two squares, so we see if it is possible to\r\n" );
document.write( "change it to the difference of two squares by adding and\r\n" );
document.write( "subtracting a square, like this\r\n" );
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document.write( "x⁴ + ___ + 4y⁴ - ___\r\n" );
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document.write( "We ask ourselves this question:\r\n" );
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document.write( "What term would have to be placed in those blanks \r\n" );
document.write( "so that the first three terms would factor as the \r\n" );
document.write( "square of the sum of their square roots, x² and 2y²?\r\n" );
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document.write( "To find out we multiply out the square of the sum of their\r\n" );
document.write( "square roots: (x² + 2y²)² = x⁴ + 4x²y² + 4y⁴\r\n" );
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document.write( "We see that the term that must be placed in the two blanks\r\n" );
document.write( "would be 4x²y², which does happen to be a square.  So we \r\n" );
document.write( "place 4x²y² in the two blanks:\r\n" );
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document.write( "x⁴ + 4x²y² + 4y⁴ - 4x²y²\r\n" );
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document.write( "Then we factor the first three terms:\r\n" );
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document.write( "(x²+2y²)² - 4x²y²\r\n" );
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document.write( "and that is the difference of two squares,\r\n" );
document.write( "and factors as\r\n" );
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document.write( "[(x²+2y²) - 2xy][(x+2y²) + 2xy]\r\n" );
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document.write( "Removing the parentheses inside the brackets:\r\n" );
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document.write( "[x² + 2y² - 2xy][x² + 2y² + 2xy]\r\n" );
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document.write( "Then we can change the brackets to parentheses\r\n" );
document.write( "and rearrange the trinomials in descending order\r\n" );
document.write( "of x and ascending order of y:\r\n" );
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document.write( "(x² - 2xy + 2y²)(x² + 2xy + 2y²)\r\n" );
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document.write( "2)What is the coefficient of x in (x⁴ + x³ + x² + x + 1)⁴?\r\n" );
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document.write( "We write the parentheses as multiplied 4 times:\r\n" );
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document.write( "(x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1)\r\n" );
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document.write( "If we were to go to the trouble to multiply that all the way out,\r\n" );
document.write( "we would have 5*5*5*5 = 625 terms before we started collecting\r\n" );
document.write( "like terms.  The only like terms in x among those 625 would be\r\n" );
document.write( "when we multiplied x from one of those 4 parenthetical expressions\r\n" );
document.write( "times 1's from the other 3.  We can choose the x any of 4 ways, so\r\n" );
document.write( "the coefficient would be 4. \r\n" );
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document.write( "3)What is the coefficient of x³ in (x⁴ + x³ + x² + x + 1)⁴?\r\n" );
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document.write( "As in the preceding problem, We write the parentheses as \r\n" );
document.write( "multiplied 4 times:\r\n" );
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document.write( "(x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1)(x⁴ + x³ + x² + x + 1)\r\n" );
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document.write( "The only terms among the 625 that would give us x³ is when we choose\r\n" );
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document.write( "1.  x³ from one of the 4 to multiply by 1's from the other three, which \r\n" );
document.write( "    we would do in 4 ways.\r\n" );
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document.write( "or\r\n" );
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document.write( "2. x² from one of the 4 to multiply by x from another one of the\r\n" );
document.write( "   other 3, and 1s from the remaining 2, which we would do in 4*3 or \r\n" );
document.write( "   12 ways.\r\n" );
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document.write( "or\r\n" );
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document.write( "3. x from 3 of them and 1 from the 4th, which we would do in 4 ways\r\n" );
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document.write( "So among the 625 multiplications there are total of 4+12+4=20 ways to get\r\n" );
document.write( "x³. So the coefficient of x³ is 20.  \r\n" );
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document.write( "Edwin
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