Zeros of Polynomials Functions

1. Zeros of Polynomials Functions How the roots, solutions, zeros, x-intercepts and factors of a polynomial function are related.

2. Polynomials A Polynomial Expression can be a monomial or a sum of monomials. The Polynomial Expressions that we are discussing today are in terms of one variable. In a Polynomial Equation, two polynomials are set equal to each other.

3. Factoring Polynomials Terms are Factors of a Polynomial if, when they are multiplied, they equal that polynomial: • (x - 3) and (x + 5) are Factors of the polynomial

4. Since Factors are a Product... …and the only way aproduct can equal zero is if one or more of the factors are zero… …then the only way the polynomial can equal zero is if one or more of the factors are zero.

5. Factor: Set each factor equal to zero and solve: Solving a Polynomial Equation Rearrange the terms to have zero on one side: The only way that x2 +2x - 15 can = 0 is if x = -5 or x = 3

6. Setting theFactorsof a Polynomial Expressionequal to zerogives theSolutionsto the Equation when the polynomial expression equals zero. Another name for the Solutions of a Polynomial is theRootsof a Polynomial ! Solutions/Roots a Polynomial

7. Zeros of a Polynomial Function A Polynomial Function is usually written in function notation or in terms of x and y. The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

8. Zeros of a Polynomial Function The Zeros of a PolynomialFunction ARE the Solutions to the PolynomialEquation when the polynomial equals zero.

9. Graph of a Polynomial Function Here is the graph of our polynomial function: The Zeros of the Polynomial are the values of x when the polynomial equals zero. In other words, the Zeros are the x-values where y equals zero.

10. x-Intercepts of a Polynomial The points where y = 0 are called the x-intercepts of the graph. The x-intercepts for our graph are the points... and (3, 0) (-5, 0)

11. x-Intercepts of a Polynomial When the Factors of a Polynomial Expression are set equal to zero, we get the Solutions or Roots of the Polynomial Equation. The Solutions/Roots of the Polynomial Equation are the x-coordinates for the x-Intercepts of the Polynomial Graph!

12. Factors, Roots, Zeros For our Polynomial Function: The Factors are: (x + 5) & (x - 3) The Roots/Solutions are: x = -5 and 3 The Zeros are at: (-5, 0) and (3, 0)

13. Remainder Theorem Let P(x) be a polynomial function. If P(x) is divided by x - c, then the remainder R = P(c).

14. Using the Remainder Theorem to Evaluate a Polynomial Book P. 282 Example 1 Find P(3) if P(x) = 2x3 – 5x2 +4x – 6 • Two Methods • Long division • Synthetic Division

15. Synthetic Division Book P. 283 Example 2 P(x) = x4 – 14x2 +5x – 9 is divided by x + 4 Synthetic Division Note Synthetic Division work ONLY very efficiently when the divisor is of the form x – c, or x + c

16. Factor Theorem 1. If f(c)=0, that is c is a zero of f, then x - c is a factor of f(x). 2. Conversely if x - c is a factor of f(x), then f(c)=0.

17. Given that is a zero of find the other zeros of f. Use the zeros to factor f. Using the Factor Theorem to Factor a Polynomial

18. Fundamental Theorem Of Algebra Every Polynomial Function with a degree higher than zero has at least one root in the set of Complex Numbers.

19. The Rational Zeros Theorem Let f be a polynomial function of degree 1 or higher of the form where each coefficient is an integer. If p/q in the lowest terms, is a rational zero of f, then p must be a factor of a0 and q must be a factor of an.

20. The Rational Zeros Theorem Find the rational zeros of Write factors of -12 and 1 to obtain the potential rational zeros.

21. Using the Rational Zero Theorem to Find the Potential Rational Zeros Book P. 287 Example 5 • Find all possible rational zeros for • f (x) = 2x3 – 3x2 – 11x + 6 • g(x) = 3x3 – 8x2 – 8x + 8

22. Find All the Rational Zeros Book P. 287 Example 6 • Find all possible rational zeros for • f (x) = 2x3 – 3x2 – 11x + 6 • g(x) = 3x3 – 8x2 – 8x + 8

23. Find All the Rational Zeros Find the rational zeros of We have already known that the possible rational zeros are:

24. Thus, -3 is a zero of f and x + 3 is a factor of f. Thus, -2 is a zero of f and x + 2 is a factor of f.

25. Thus f(x) factors as:

27. 3.3 The Theory of Equations The Multiplicity of a Root/Zero of Polynomials: If the factor x – c occurs k times in the complete factorization of the polynomial P(x), then c is called a root/zero of P(x) = 0 with multiplicity k

28. The Conjugate Pairs Theorem If P(x) = 0 is a polynomial equation with real coefficients and the complex number a + bi (b0) is a root/zero, the a – bi is also a root/zero.

29. Find Roots/Zeros of a Polynomial We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring. Some are easier to factor than others! The roots are: 0, -2, 2

30. (x - 5) is a factor Find Roots/Zeros of a Polynomial If we cannot factor the polynomial, but know one of the roots, we can divide that factor into the polynomial. The resulting polynomial has a lower degree and might be easier to factor or solve with the quadratic formula. We can solve the resulting polynomial to get the other 2 roots:

31. Complex Conjugates Theorem Roots/Zeros that are not Real are Complex with an Imaginary component. Complex roots with Imaginary components always exist in Conjugate Pairs. If a + bi (b ≠ 0) is a zero of a polynomial function, then its Conjugate, a - bi, is also a zero of the function.

32. Ex: Find all the roots of If one root is 4 - i. Find Roots/Zeros of a Polynomial If the known root is imaginary, we can use the Complex Conjugates Pairs Theorem. Because of the Complex Conjugate Theorem, we know that another root must be 4 + i. Can the third root also be imaginary? No. Because the degree of f(x) is 3 which is an odd number, if we have the third root/zero is imaginary, then by the Conjugate Pairs Theorem, we must have the fourth root (the conjugate of the third one). Then the degree of f(x) is 4 not 3. This is a contradiction.

33. Ex: Find all the roots of If one root is 4 - i. Example (con’t) If one root is 4 - i, then one factor is [x - (4 - i)], and Another root is 4 + i, & another factor is [x - (4 + i)]. Multiply these factors:

34. If the product of the two non-real factors is Ex: Find all the roots of then the third factor (that gives us the neg. real root) is the quotient of P(x) divided by : If one root is 4 - i. Example (con’t) The third root is x = -3

35. Now write a polynomial function of least degree that has real coefficients, a leading coeff. of 1 and 1, -2+i, -2-i as zeros. • f(x)= (x-1)(x-(-2+i))(x-(-2-i)) • f(x)= (x-1)(x+2 - i)(x+2+ i) • f(x)= (x-1)[(x+2) - i] [(x+2)+i] • f(x)= (x-1)[(x+2)2 - i2] Foil • f(x)=(x-1)(x2 + 4x + 4 – (-1)) Take care of i2 • f(x)= (x-1)(x2 + 4x + 4 + 1) • f(x)= (x-1)(x2 + 4x + 5) Multiply • f(x)= x3 + 4x2 + 5x – x2 – 4x – 5 • f(x)= x3 + 3x2 + x - 5

36. Now write a polynomial function of least degree that has real coefficients, a leading coeff. of 1 and 4, 4, 2+i as zeros. • Note: 2+i means 2 – i is also a zero • F(x)= (x-4)(x-4)(x-(2+i))(x-(2-i)) • F(x)= (x-4)(x-4)(x-2-i)(x-2+i) • F(x)= (x2 – 8x +16)[(x-2) – i][(x-2)+i] • F(x)= (x2 – 8x +16)[(x-2)2 – i2] • F(x)= (x2 – 8x +16)(x2 – 4x + 4 – (– 1)) • F(x)= (x2 – 8x +16)(x2 – 4x + 5) • F(x)= x4– 4x3+5x2 – 8x3+32x2 – 40x+16x2 – 64x+80 • F(x)= x4-12x3+53x2-104x+80