<p> Further Concepts for Advanced Mathematics - FP1</p><p>Unit 4 Identities and Roots of Equations</p><p>4c Cubic, Quartic and Quintic Equations</p><p>Cubic Equations</p><p>The three roots of the cubic equation ax 3  bx 2  cx  d  0 are called  ,  and  (alpha, beta and gamma). The solutions to the equation are x   , x   and x   .</p><p>To find the relationships between the coefficients in the original equation and the roots, we have to use a different technique.</p><p>Since the solutions to the equation are x   , x   and x   , the equation must have the factors (x ) , (x   ) and (x   ) . Multiplying these together would give the first term x 3 rather than ax 3 . It follows that the actual factorisation of ax 3  bx 2  cx  d is a(x )(x   )(x   )</p><p>This gives us the identity</p><p> ax 3  bx 2  cx  d  a(x )(x   )(x   ) multiplying out the right-hand side gives </p><p> ax 3  bx 2  cx  d  a(x )(x 2  x  x   ) ax 3  bx 2  cx  d  a(x 3  x 2  x 2  x x 2  x  x  ) ax 3  bx 2  cx  d  ax3  a(     )x 2  a(     )x  a</p><p>Equating coefficients of x 2 gives b  a(     )</p><p> b Hence        the sum of the roots a</p><p>Equating coefficients of x gives c  a(     )</p><p> c Hence       the sum of the pairs of roots a</p><p>Equating the constant terms gives d  a</p><p> d Hence    the product of the roots a</p><p>You should notice that there are some similarities between these formulae for cubic equations and those for quadratic equations. There is a formula for the sum and product just like the quadratic equations but there is also a formula for the sum of all the 1 possible different pairings of the roots. You will find that a new formula appears every time we increase the order of the equation we look at. For a quartic equation, there will be four formulae; the sum of the roots, the sum of the pairs of the roots, the sum of the triples of the roots and the product of the roots.</p><p>Notation</p><p>To cope with the confusion that could arise, the following notation is used</p><p> is used to stand for the sum of the roots</p><p>For a cubic equation,        For a quartic equation,         </p><p> is used to stand for the sum of the pairs of roots</p><p>For a cubic equation,        For a quartic equation,             </p><p> is used to stand for the sum of the triples of roots</p><p>For a quartic equation,         </p><p>Examples with the roots of cubic equations</p><p>1. The roots of the cubic equation 2x3  3x 2  4x  5  0 are  ,  and  .</p><p>Find the cubic equations with roots i)   2 ,   2 and   2 1 1 1 ii) , and   </p><p>For the original equation </p><p> 3 b   the sum of the roots is   2 a</p><p>4 c    2 the sum of the pairs of the roots is  2 a</p><p>5 d    the product of the roots is  2 a</p><p> i) The sum of the new roots is   2    2    2        6</p><p>2 3 21 This is    6    6    2 2</p><p>The sum of the pairs of new roots is (  2)(  2)  (  2)(  2)  (  2)(  2)</p><p> which simplifies to</p><p>  2  2  4    2  2  4    2  2  4        4(     ) 12    4 12 3  2  4  12 2  2  6 12  20</p><p>The product of the new roots is (  2)(  2)(  2)</p><p> which simplifies to</p><p>(  2)(  2  2  4)    2  2  4  2  4  4  8    2  4  8 5 3    2 2  4   8 2 2 5    4  6  8 2 41   2</p><p>So the new equation is</p><p> 21  41 x3   x 2  20x     0  2   2  21 41 x3  x 2  20x   0 2 2 2x3  21x 2  20x  41  0</p><p>1 1 1       2 4         ii) The sum of the new roots is 5       2 5</p><p>1 1 1 1 1 1 1 1 1 The sum of the pairs of new roots is                 </p><p>Using a common denominator, this simplifies to</p><p>3 1 1 1                  3  3  2  5  2  5</p><p>1 1 1 1 2 The product of the new roots is          5</p><p>The new equation is</p><p> 4  3  2  x3   x 2  x     0  5  5  5  4 3 2 x3  x 2  x   0 5 5 5 5x 3  4x 2  3x  2  0</p><p>Compare this to the original equation 2x3  3x 2  4x  5  0</p><p>Do you notice anything?</p><p>Did this happen for quadratic equations (look back at the examples on 4b)</p><p>Can you make a generalisation about making a new equation using the reciprocals of the original roots?</p><p>Quartic and Quintic Equations</p><p>Rules for the roots of quartic and quintic equations can be found by following the same ideas as for quadratic and cubic equations.</p><p>For the quartic equation ax 4  bx3  cx 2  dx  e  0 , the roots are called  ,  ,  and  (alpha, beta, gamma and delta). The solutions to the equation are x   , x   , x   and x   .</p><p>For any quartic equation: b     a c   the sum of the pairs of roots i.e.             a d    this is the sum of the combinations of 3 roots i.e.         a e   a</p><p>4 For the quintic equation ax 5  bx 4  cx3  dx 2  ex  f  0 , the roots are called  ,  ,  ,  and (alpha, beta, gamma, delta and epsilon). The solutions to the equation are x   , x   , x   , x   and x   .</p><p>For any quartic equation: b     a c   the sum of the pairs of roots i.e.                     a d    this sum of the triples (work these out for yourself!)  a e   the sum of the combinations of 4 roots  a f    a</p><p>This can make working with equations of higher order difficult but there is a quick method for finding some of the new equations that are based on the roots of an original equation.</p><p>The Substitution Method</p><p>This method can be used to find new equations where the same thing is being done to each of the original roots.</p><p>E.G. The roots of 2z 3  3z 2  5z 1  0 are  ,  and  .</p><p>Find the cubic equation with roots 3  2 ,3  2 and3  2</p><p>If the original equation is written in terms of z , we can write a new equation in terms of another variable w , where w  3  2z .</p><p>3  w If w  3  2z then z  . 2</p><p>We can substitute this into the original equation giving</p><p>3 2  3  w   3  w   3  w  2   3   5  1  0  2   2   2  27  27w  9w2  w3 9  6w  w2 3  w 2  3  5 1  0 8 4 2 27  27w  9w2  w3 27 18w  3w2 5w 15   1  0 4 4 2 27  27w  9w2  w3  27 18w  3w2 10w  30  4  0  w3 12w2  35w  28  0 w3 12w2  35w  28  0</p><p>5 6</p>