1. Objectives
Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots.
Identify all of the roots of a polynomial equation.

2. You have learned several important properties about real roots of polynomial equations.
You can use this information to write polynomial function when given in zeros.

3. Check It Out! Example 1a
Write the simplest polynomial function with the given zeros.
–2, 2, 4
If r is a zero of P(x), then x – r is a factor of P(x).
P(x) = (x + 2)(x – 2)(x – 4)
P(x) = (x2 – 4)(x – 4)
Multiply the first two binomials.
P(x) = x3– 4x2– 4x + 16
Multiply the trinomial by the binomial.

4. Notice that the degree of the function in Example 1 is the same as the number of zeros. This is true for all polynomial functions. However, all of the zeros are not necessarily real zeros. Polynomials functions, like quadratic functions, may have complex zeros that are not real numbers.

5. Using this theorem, you can write any polynomial function in factor form.
To find all roots of a polynomial equation, you can use a combination of the Rational Root Theorem, the Irrational Root Theorem, and methods for finding complex roots, such as the quadratic formula.

6. Example 2: Finding All Roots of a Polynomial
Solve x4 – 3x3 + 5x2 – 27x – 36 = 0 by finding all roots.
The polynomial is of degree 4, so there are exactly four roots for the equation.
Step 1 Use the rational Root Theorem to identify rational roots.
±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36
p = –36, and q = 1.

7. Example 2 Continued
Step 2 Graph y = x4 – 3x3 + 5x2 – 27x – 36 to find the real roots.
Find the real roots at or near –1 and 4.

8. Example 2 Continued
Step 3 Test the possible real roots.
Test –1. The remainder is 0, so (x + 1) is a factor. –1
1 – –27 –36 –1 4 –9 36 1 –4 9
–36

9. Example 2 Continued
The polynomial factors into (x + 1)(x3 – 4x2 + 9x – 36) = 0. 4
1 – –36
Test 4 in the cubic polynomial. The remainder is 0, so (x – 4) is a factor. 4 36 1 9

10. Example 2 Continued
The polynomial factors into (x + 1)(x – 4)(x2 + 9) = 0.
Step 4 Solve x2 + 9 = 0 to find the remaining roots.
x2 + 9 = 0
x2 = –9
x = ±3i
The fully factored form of the equation is (x + 1)(x – 4)(x + 3i)(x – 3i) = 0. The solutions are 4, –1, 3i, –3i.

11. Example 3: Writing a Polynomial Function with Complex Zeros
Write the simplest function with zeros 2 + i, , and 1.
Step 1 Identify all roots.
By the Rational Root Theorem and the Complex Conjugate Root Theorem, the irrational roots and complex come in conjugate pairs. There are five roots: 2 + i, 2 – i, , , and 1. The polynomial must have degree 5.

12. Example 3 Continued
Step 2 Write the equation in factored form.
P(x) = [x – (2 + i)][x – (2 – i)](x – )[(x – ( )](x – 1)
Step 3 Multiply.
P(x) = (x2 – 4x + 5)(x2 – 3)(x – 1)
= (x4 – 4x3+ 2x2 + 12x – 15)(x – 1)
P(x) = x5 – 5x4+ 6x3 + 10x2 – 27x – 15

13. Lesson Quiz: Part I
Write the simplest polynomial function with the given zeros.
1. 2, –1, 1
2. 0, –2,
3. 2i, 1, –2
4. Solve by finding all roots x4 – 5x3 + 7x2 – 5x + 6 = 0
x3 – 2x2 – x + 2
x4 + 2x3 – 3x2 – 6x
x4 + x3 + 2x2 + 4x – 8
2, 3, i,–i