Mathematical Investigations III
Name:
Mathematical Investigations III - A View of the World
Sum/Product of Roots
1. Let f(x) = x2 – 3x – 10 and let r1 and r2 be the zeroes (roots) of f(x).
(a) Solve the equation f ( x)  0 to find r1 and r2.
(b) Find the sum of the roots, r1  r2 .
(c) Find the product of the roots, r1  r2 .
(d) What is the relationship between your answers to (b) and (c) and the coefficients of
f (x ) ?
Let's look at the relationships between the roots of a polynomial and its coefficients.


  
Let P2 x  x  r1 x  r2 . Then the roots of P ( x) are r and r .
Let P1 x  x  r1 . P1 has a single root r1.
2
1
2
In expanded form, we have P2 ( x)  x  (r1  r2 ) x  r1r2 , or
P2 ( x)  x 2  (sum of roots) x   product of roots 
2
 



2. Let P3 x  x  r1 x  r2 x  r3 . Expand P3 ( x) and combine like terms as above.
 




3. Repeat this process with P4 x  x  r1 x  r2 x  r3 x  r4 . (Note: Wolfram|Alpha will
do this for you if you just input P4 ( x) in factored form and ask it to “expand”.)
Poly 10.1
Rev.S11
Mathematical Investigations III
Name:
4. Let f ( x)  ( x  1)5 .
(a) Expand f (x) .
(b) What are the sum and product of the roots of f (x) ?
5. Let g ( x)  ( x  1)6 .
(a) Expand g (x ) .
(b) What are the sum and product of the roots of g (x ) ?
6.
Reflecting on your answers to the last four questions, describe how you can use the
coefficients of a polynomial to determine the sum and product of its roots? Be precise.
Apply what you have observed above:
7.
Given each of the following polynomials, find both the sum and the product of the roots.
2
3
2
(a) f (x)  x  12x  25
(b) g(x)  x  4x  2x  20
8.
Find the sum and products of the solutions to the following polynomial equations.
4
3
2
(a) x  5x  x  6x  3 = 0
3
(b) x  12x  28  0
Poly 10.2
(Caution)
Rev.S11
Mathematical Investigations III
Name:
2
9. The sum of the solutions of the polynomial equation 2x  10x  28  0 is NOT 10 and
the product is NOT –28.
(a) Solve the polynomial equation above to find its actual roots. What is their
sum? What is their product?
(b) Why is the sum NOT 10 and the product NOT -28?
(c) Explain what must be done to find the sum and product of the roots when the
leading coefficient is not 1.
10. Find the sum and product of the solutions of:
2
(a) 6x  20x  16  0
3
2
(b) 2x  3x  6x  9  0
11. The polynomial p( x)  x3  4 x 2  7 x  10 has roots at x  1 and x  2 . Where is the third
root? How do you know?
12. The polynomial w( x)  2 x3  17 x 2  80 x  125 has a root at x  3  4i . From previous
experience you should be able to write down a second root. Then using the ideas in this
packet, you should be able to find the third root. What are these two roots? Explain how
you found them.
Poly 10.3
Rev.S11
Mathematical Investigations III
Name:
Extra practice.
13.
Using the pattern you observed in Question 6, find a quadratic polynomial function with
integer coefficients whose zeros are 3  5 and 3  5 . Write your answer in the
form f ( x)  ax 2  bx  c .
14. Stu Dent solved a quadratic equation with integer coefficients by factoring and setting each
factor equal to zero. His answers are: x  2 and x   4 . Use the pattern you found in
3
3
Question 6 to find a possible equation that Stu could have solved.
15. Without actually solving for x, find the sum of the squares of the solutions of the
3
2
polynomial equation x  9x  12x  14  0 . (Hint: Look back to Question 2.)
16. The polynomial q( x)  x3  5 x 2  9 x  45 has only one real root, namely at x  5 . Without
doing any sort of computation, explain why the other two roots of q(x) are purely imaginary
(that is, bi for some real number b).
Poly 10.4
Rev.S11
Mathematical Investigations III
Name:
Poly 10.5
Rev.S11