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6-6. Fundamental Theorem of Algebra. Warm Up. Lesson Presentation. Lesson Quiz. Holt Algebra 2. Warm Up Identify all the real roots of each equation. 1. 4 x 5 – 8 x 4 – 32 x 3 = 0. 0, –2, 4. 1, –3, 3. 2. x 3 – x 2 + 9 = 9 x. –1, 1, –4, 4. 3. x 4 + 16 = 17 x 2.

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6-6

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  1. 6-6 Fundamental Theorem of Algebra Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

  2. Warm Up Identify all the real roots of each equation. 1. 4x5 – 8x4 – 32x3 = 0 0, –2, 4 1, –3, 3 2. x3 –x2 + 9 = 9x –1, 1, –4, 4 3. x4 + 16 = 17x2 4. x3 + 5x2– 3x – 3 = 0.

  3. 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.

  4. 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.

  5. 2 3 P(x) = (x + 1)(x – )(x – 4) 2 2 11 1 8 3 3 3 3 3 P(x) = (x2+ x – )(x – 4) P(x) = x3 – x2 – 2x + Example 1: Writing Polynomial Functions Write the simplest polynomial with roots –1, , and 4. If r is a zero of P(x), then x – r is a factor of P(x). Multiply the first two binomials. Multiply the trinomial by the binomial.

  6. 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. Multiply the trinomial by the binomial. P(x) = x3– 4x2– 4x + 16

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. Example 2 Continued Step 3 Test the possible real roots. Test –1. The remainder is 0, so (x + 1) is a factor. –1 1 –3 5 –27 –36 –1 4 –9 36 1 –4 9 –36 0

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

  13. 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.

  14. Check It Out! Example 2 Solve x4 + 4x3 – x2+16x – 20 = 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, ±4, ±5, ±10, ±20 p = –20, and q = 1.

  15. Check It Out! Example 2 Continued Step 2 Graph y = x4 + 4x3 – x2+ 16x – 20 to find the real roots. Find the real roots at or near –5 and 1.

  16. Check It Out! Example 2 Continued Step 3 Test the possible real roots. Test 1. The remainder is 0, so (x - 1) is a factor. 1 1 4 –1 16 –20 1 5 4 20 1 5 4 20 0

  17. Example 3: Writing a Polynomial Function with Complex Zeros Write the simplest function with zeros 2 + i, , and 1. Step 1Identify 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.

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

  19. Check It Out! Example 3 Write the simplest function with zeros 2i, and -3 Step 1Identify all roots. By the Rational Root Theorem and the Complex Conjugate Root Theorem, the irrational roots and complex come in conjugate pairs. There are three roots: 2i, –2i, and -3. The polynomial must have degree 3.

  20. Check It Out! Example 3 Continued Step 2Write the equation in factored form. P(x) = [ x - (2i)][x + (2i)][x + 3] Step 3Multiply. P(x) = x3+ 3x2 + 2x + 6

  21. 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

  22. 4 3 Lesson Quiz: Part II The volume of a cylindrical vitamin pill with a hemispherical top and bottom can be modeled by the function V(x) = 10r2 + r3, where r is the radius in millimeters. For what value of r does the vitamin have a volume of 160 mm3? 5. about 2 mm

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