Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5

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1 Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares: Note: There is no such thing sum of squares! Sum of Cubes: Difference of Cubes: Verify these 2 patterns by multiplying: 3 Example - Factor. a. x 3-8 b. 8x 3-1 c s 3 d. 64x e. 216x 3-1 f. 8x

2 4 Example - Solve. Find all complex roots. a. 27x = 0 b. 8x Example - Factor. (Hint: a quadratic in disguise...) a. x 4-2x 2-8 b. x 4 + 7x Example - Solve the higher degree polynomial equation. a. x 4 - x 2 = 12 b. x x = 0

3 Quick Check Factor. 1. x 4-5x x 4-3x 2-10 Solve the equation. (See example 4) 3. 27x 3-1 = 0 4. x = 0 5. x 4 + 3x 2-28 = = -x 4-12x 2

4 Day 7: 6.5 Theorems about Roots of Polynomial Equations Warm Up List all the integer factors of each number Multiply. 4. (x - 5)(x 2 + 7) 5. (x + 2)(x + 3)(x - 3) Recall rational numbers can be written as a Their decimal representations or 8. irrational numbers cannot be written as a quotient of integers. Their decimal representations do not repeat and do not terminate. 9. imaginary numbers are non-real numbers and written in the form of 10. root is also known as an or a (graphically) (algebraically) I. Rational Root Theorem If is in simplest form and is a rational root of the polynomial equation... with integer coefficients, then p must be a factor of a 0, and q must be a factor of a n. What does this mean in English? 1 Example - Find the rational roots. x 3 + x 2-3x - 3 = 0 Step 1: List the possible rational roots. Step 2: Test each of them.

5 2 Example - Use the rational root theorem to find all roots of the equation. 2x 3 - x 2 + 2x - 1 = 0 Step 1: List the possible rational roots. Step 2: Test each of them. Step 3: Use synthetic division with the root you found in step 2 to find the quotient. Step 4: Find the roots of the quotient. Quick Check. For 1-2, find the rational roots. 1. x 3-4x 2-2x + 8 = 0 2. x 3-2x 2-5x + 10 = x 3 + x 2 - x + 1 = 0 (Find all roots)

6 Recall: Solve using the quadratic formula. x 2-4x - 1 = 0 II. Irrational Roots In problem above, the 2 solutions are and 2-5. This pair is called... Conjugates - are solutions to polynomial equations. They ALWAYS come in pairs of the form a + b and a - b. Therefore, if you find one conjugate, you automatically know the other conjugate. Irrational Root Theorem Let a and b be rational numbers and b be an irrational number. If a + b is a root (solution) or a polynomial equation with rational coefficients, then the conjugate a - b is also a root. Notice that you conjugate the b. 3 Example (Irrational Roots) Given are roots of a polynomial equation with integer coefficients. Find two additional roots. a and - 11 b. 2-7 and 5 III. Imaginary Roots Complex Conjugates - number pairs of the form a + bi and a - bi. You can use complex conjugates to find an equation s imaginary roots. Imaginary Root Theorem - If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a - bi also is a root. Notice that you conjugate bi. 4 Example (Imaginary Roots) Given are roots of a polynomial equation with integer coefficients. Find two additional roots. a. 3 - i and 2i b. 3i and -2 + i

7 5 Example Find a third degree polynomial equation with rational coefficients that has roots 3 and 1 + i. Step 1: Find the other root using Imaginary Root Theorem. Step 2: Write the polynomial in factored form using Factor Theorem. Step 3: Multiply the factors to write the polynomial in standard form. Quick Check Given are roots of a polynomial equation with integer coefficients. Find two additional roots. a. 2-5 and 7 b. 3i and -2 + i c. Find a fourth degree polynomial equation with rational coefficients that has roots i and 2i. Day Fundamental Theorem of Algebra Warm Up State the degree of each polynomial. 1. 3x 2 - x x x x Solve each equation using the quadratic formula. 4. x = 0 5. x 2-2x + 3 = x 2 + 5x + 4 = 0

8 I. The Fundamental Theorem of Algebra Activity: Zeros a. Find the solutions. b. Are the solutions real or imaginary? c. How many solutions are there? 1. x 4-5x = 0 2. x 4 + 7x = 0 3. Make a conjecture about the number of zeros of a fourth degree polynomial, regardless of the types of zeros. A fourth degree polynomial function has 4 zeros regardless of type. All along we have been solving polynomial equations; we have been finding their roots. The roots have turned out to be integers, rational, irrational, and imaginary. In other words, roots can be complex numbers. Diagram of numbers: Fundamental Theorem of Algebra (FTA) Corollary Including complex roots and multiple roots, an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros. In other words, you can factor a polynomial of degree n into n linear factors. The number n includes multiple roots. For example, x 3 = 0 has roots and can be written as or (linear factor form) The equation has 3 same linear factors. The 3 roots are the same, - they are all.

9 1 Example Find the number of complex roots, the possible number of real roots, and the possible rational roots. a. x 3 + 2x 2-4x - 6 = 0 By Fundamental Theorem of Algebra, there are complex roots. By Imaginary Root Theorem, there are either imaginary roots or imaginary roots ( conjugate pair). So there are either real roots or real root. By the Rational Root Theorem, the possible rational roots are,,, and. b. x 4-3x 3 + x 2 - x + 3 = 0 2 Example (Must know!) State the number of complex zeros. Then find all the zeros. (Note: You often have to use a combination of graphing, the Factor Theorem, polynomial division, the Remainder Theorem, and the Quadratic Formula!) a. f(x) = x 3 + x 2 - x + 2

10 b. f(x) = x 3-2x 2 + 4x - 8 Day Fundamental Theorem of Algebra (continued) Warm Up 1. Find the number of complex roots, the possible number of real roots, and the possible rational roots. x 4-3x 3 + 4x + 1 = 0 Complex roots: Possible number of real roots: Possible rational roots: 2. State the number of complex roots. Then find all the zeros. f(x) = x 5 + 3x 4 - x - 3 Day 10 Review Chapter 6 Day 11 Test Chapter 6