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Determine the zeros of

+2 votes

f(x) = x^4 – x^3 + 7x^2 – 9x – 18

asked Feb 1, 2013 in ALGEBRA 2 by linda Scholar

3 Answers

+2 votes

f(x) = x4 - x3 + 7x2 - 9x -18

x4 - x3 + 7x2 - 9x -18 = 0

Synthetic division theorem, possible roots are ±1, ±2, and ±3.

    -1 | 1    -1    7     -9     -18

        |  0    -1    2     -9      18

        |_______________________

     2 | 1     -2    9    -18     0

        | 0      2    0      18

        |_______________________

         1      0     9      0

By checking , we find that -1 and 2 are a roots.

By the fundamental theorem of algebra.

The remaining root are those (x2 + 9) = 0

Subtract 9 from each side.

x2 = - 9

Apply square root each side.

x = √(-9) = √(9i2) = ± 3i                   [i2 = -1]

So the roots are  -1, 2, 3i and -3i.

answered Feb 1, 2013 by richardson Scholar
Thank you!!! How would you know the possible roots? any rule to be applied?
0 votes

Identify Rational Zeros :

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation anxn + an  1xn – 1 + ... + a1x + a0 = 0, then p is a factor of a0 and q is a factor if an.

The function f (x) = x4 - x3 + 7x2 - 9x - 18.

If p/q is a rational zero, then p is a factor of 18 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, ± 9.

The possible values for q are ± 1

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 9.

Make a table for the synthetic division and test possible real zeros.

p/q

   1

  - 1

7

- 9

18

1

1

   0

7

- 2

 -16

- 1

1

- 2

 9 

- 18

0

Since f(- 1) = 0, x = – 1 is a zero. The depressed polynomial is  x3 - 2x2 + 9x - 18 = 0.

answered Jun 26, 2014 by lilly Expert
0 votes

Contd.............

If p/q is a rational zero, then p is a factor of 3 and q is a factor of 1.

By the Rational Roots Theorem, the only possible rational roots are, p/q = ± 1, ± 2, ± 9.

Make a table for the synthetic division and test possible real zeros.

p/q

1

- 2

9

- 18

1

1

- 1

8

- 10

- 1

1

- 3

12

- 30

2

1

0

9

0

Since f(2) = 0, x = 2 is a zero. The depressed polynomial is  x2 + 9 = 0.

The depressed polynomial is  x2 + 9 = 0.

 x2 = - 9

 x2 = (± 3i)2

x = - 3i and x = 3i.

The function has two real zeros at x = -1 and x = 2, two imaginary zeros at x = - 3i and 

x = 3i.

answered Jun 26, 2014 by lilly Expert

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