1. Introduction Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial function. This information allows for more accurate sketches of functions. 1 2.3.3: Finding Zeros

2. Key Concepts Recall that roots are the x-intercepts of a function. In other words, these are the x-values for which a function equals 0. These zeros are another way of referring to the roots of a function. When a polynomial equation with a degree greater than 0 is solved, it may have one or more real solutions, or it may have no real solutions (in which case it would have complex solutions). 2 2.3.3: Finding Zeros

3. Key Concepts Recall that both real and imaginary numbers belong to the set of complex numbers; therefore, all polynomial functions with a degree greater than 0 will have at least one root in the set of complex numbers. This is referred to as the Fundamental Theorem of Algebra. 3 2.3.3: Finding Zeros Fundamental Theorem of Algebra If p(x) is a polynomial function of degree n ≥ 1 with complex coefficients, then the related equation p(x) = 0 has at least one complex solution (root).

4. Key Concepts, continued A repeated root is a root that occurs more than once in a polynomial function. Recall that the solutions to a quadratic equation that contains imaginary numbers come in pairs. These are called complex conjugates, the complex number that when multiplied by another complex number produces a value that is wholly real; the complex conjugate of a + bi is a – bi. 4 2.3.3: Finding Zeros

5. Key Concepts, continued If an imaginary number is a zero of a function, its conjugate is also a zero of that function. This is true for all polynomial functions, and is known as the Complex Conjugate Theorem. 5 2.3.3: Finding Zeros Complex Conjugate Theorem Let p(x) be a polynomial with real coefficients. If a + bi is a root of the equation p(x) = 0, where a and b are real and b ≠ 0, then a – bi is also a root of the equation.

6. Key Concepts, continued For a polynomial function p(x), the factor of a polynomial is any polynomial that divides evenly into that function. Recall that when a polynomial is divided by one of its factors, there is a remainder of 0 and the result is a depressed polynomial. This is an illustration of the Factor Theorem. 6 2.3.3: Finding Zeros Factor Theorem The binomial x – a is a factor of the polynomial p(x) if and only if p(a) = 0, where a is a real number.

7. Key Concepts, continued The Factor Theorem can help to find all the factors of a polynomial. To do this, first show that the binomial in question is a factor of the polynomial. If the remainder is 0, then the binomial is a factor. Then, determine if the resulting depressed polynomial can be factored. The identified factors indicate where the function crosses the x-axis. The zeros of a function are related to the factors of the polynomial. The graph of a polynomial function shows the zeros of the function, which are the x-intercepts of the graph. 7 2.3.3: Finding Zeros

8. Key Concepts, continued It is often helpful to know which integer values of a to try when determining p(a) = 0. Use the Integral Zero Theorem to determine the zeros of a polynomial function. Identify the factors of the constant term of a polynomial function and use substitution to determine if each number results in a zero. Use synthetic division to determine the remaining factors. 8 2.3.3: Finding Zeros

9. Key Concepts, continued For example, consider the equation p(x) = x 2 + 10x + 25. a n = 1 and a 0 = 25. A possible zero of this function is –5 because –5 is a factor of 25. The zeros of any polynomial function correspond to the x-intercepts of the graph and to the roots of the corresponding equation. 9 2.3.3: Finding Zeros Integral Zero Theorem If the coefficients of a polynomial function are integers such that a n = 1 and a 0 ≠ 0, then any rational zeros of the function must be factors of a 0.

10. Key Concepts, continued If a polynomial has a factor x – a that is repeated n times, then the root is called a repeated root and x = a is a zero of multiplicity. Multiplicity refers to the number of times a zero of a polynomial function occurs. If the multiplicity is odd, then the graph intersects the x-axis at the point (x, 0). If the multiplicity is even, then the graph just touches the axis at the point (x, 0). 10 2.3.3: Finding Zeros

11. Key Concepts, continued If p(x) is a polynomial with real coefficients whose terms are arranged in descending powers of the variable, then the number of positive zeros of y = p(x) is the same as the number of sign changes of the coefficients of the terms, or is less than this by an even number. Recall that when solving a quadratic function, we used the quadratic formula, which generated pairs of roots. Often, these roots were complex and the x-intercepts could not be graphed on the coordinate plane. For this reason, we must count down from our maximum number of zeros by twos to determine the number of real zeros. For example, for a polynomial with 3 sign changes, the number of positive zeros may be 3 or 1, since 3 – 2 = 1. 11 2.3.3: Finding Zeros

12. Key Concepts, continued Also, the number of negative zeros of y = p(x) is the same as the number of sign changes of the coefficients of the terms of p(–x), or is less than this number by an even number. 12 2.3.3: Finding Zeros

13. Common Errors/Misconceptions not finding all the factors of a polynomial function making sign errors when performing synthetic division misusing the terms “roots” and “factors” 13 2.3.3: Finding Zeros

14. Guided Practice Example 1 Given the equation x 3 + 4x 2 – 3x – 18 = 0, state the number and type of roots of the equation if one root is –3. 14 2.3.3: Finding Zeros

15. Guided Practice: Example 1, continued 1.Use synthetic division to find the depressed polynomial. The related polynomial is x 3 + 4x 2 – 3x – 18, with coefficients 1, 4, –3, and –18. One root of the equation is –3; therefore, a factor of the related polynomial is [x – (–3)], which simplifies to x + 3. 15 2.3.3: Finding Zeros

16. Guided Practice: Example 1, continued Divide the related polynomial by x + 3 to find the depressed polynomial. Let the value of a be –3. The depressed polynomial is x 2 + x – 6. 16 2.3.3: Finding Zeros

17. Guided Practice: Example 1, continued 2.Factor the depressed polynomial to find the remaining factors. Use previously learned strategies to factor the polynomial. x 2 + x – 6 Depressed polynomial (x – 2)(x + 3) Factor the polynomial. The remaining factors are (x – 2) and (x + 3). 17 2.3.3: Finding Zeros

18. Guided Practice: Example 1, continued 3.State the number of roots and type of roots of the equation. The factors of the related polynomial are (x + 3), (x – 2), and (x + 3). Recall that we divided the depressed polynomial by (x + 3) before finding the remaining factors, so we must include (x + 3) as a factor. Therefore, the roots of the equation are –3, –3, and 2. Since –3 appears twice, it is a repeated root. Because of the repeated root, this equation has only 2 real roots: –3 and 2. 18 2.3.3: Finding Zeros ✔

19. Guided Practice: Example 1, continued 19 2.3.3: Finding Zeros

20. Guided Practice Example 3 Write the simplest polynomial function with integral coefficients that has the zeros 5 and 3 – i. 20 2.3.3: Finding Zeros

21. Guided Practice: Example 3, continued 1.Determine additional zeros of the function. Because 3 – i is a zero, then according to the Complex Conjugate Theorem, the conjugate 3 + i is also a zero. 21 2.3.3: Finding Zeros

22. Guided Practice: Example 3, continued 2.Use the zeros to write the polynomial as a product of the factors. The zeros are 5, 3 – i, and 3 + i. These can be written as the factors x – 5, x – (3 – i), and x – (3 + i). The polynomial function written in factored form with zeros 5 and 3 – i is f(x) = (x – 5)[x – (3 – i )][x – (3 + i )]. 22 2.3.3: Finding Zeros

23. Guided Practice: Example 3, continued 3.Multiply the factors to determine the polynomial function. 23 2.3.3: Finding Zeros f(x) = (x – 5)[x – (3 – i )][x – (3 + i )] Polynomial function written in factored form f(x) = (x – 5)[(x – 3) – i ][(x – 3) + i ]Regroup terms. f(x) = (x – 5)[(x – 3) 2 – i 2 ] Rewrite as the difference of two squares. f(x) = (x – 5)(x 2 – 6x + 9 – i 2 )Simplify.

24. Guided Practice: Example 3, continued The polynomial function of least degree with integral coefficients whose zeros are 5 and 3 – i is f(x) = x 3 – 11x 2 + 40x – 50. 24 2.3.3: Finding Zeros f(x) = (x – 5)[x 2 – 6x + 9 – (–1)] Replace i 2 with –1, since i 2 = –1. f(x) = (x – 5)(x 2 – 6x + 9 + 1)Simplify. f(x) = x 3 – 11x 2 + 40x – 50Distribute. ✔

25. Guided Practice: Example 3, continued 25 2.3.3: Finding Zeros