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Topic #6: Polynomial Equations

Topic #6: Polynomial Equations. Drill. Find the zeros of the following function (in factored form) Find a quadratic equation (in polynomial form) with zeros at -3, and 4. Factor the following polynomial equation, then identify the zeros:. Topic #6 Objectives.

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Topic #6: Polynomial Equations

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  1. Topic #6: Polynomial Equations

  2. Drill • Find the zeros of the following function (in factored form) • Find a quadratic equation (in polynomial form) with zeros at -3, and 4. • Factor the following polynomial equation, then identify the zeros:

  3. Topic #6 Objectives • After completing the topic Polynomial equations, students will be able to • define and use imaginary and complex numbers in the solution of quadratic equations; • use the discriminant of a quadratic equation to determine the number and type of roots of the equation; • understand the implications of the Fundamental Theorem of Algebra and the Remainder Theorem, and solve quadratic and higher degree polynomial equations.

  4. Vocabulary for Topic #6 • quadratic formula, • imaginary numbers, • complex numbers, • discriminant, • real roots, and • complex roots Extra Credit vocabulary due Friday!

  5. Drill Solve each equation.

  6. Drill Solve the following equations. (Factor and find the zeros)

  7. Drill Solve the following equations.

  8. Drill Find the vertex (min/max) and zeros of the following function: 1. Solve the following equations: (you may round answers to the nearest hundreth) 2. 3.

  9. Word Problem Practice A company plans to build a large movie theatre. The financial analyst determined that the profit function for the theatre was where x is the number of movie theatres and P(x) is the profit earned in thousands of dollars. Determine the range of numbers of movie screens that will guarantee a profit.

  10. Word Problem Practice #2 During practice, a softball pitcher throws a ball whose height can be modeled by the equation h = -16t 2+ 24t +1, where h = height in feet and t = time in seconds. At what time(s) does the ball reach a height of 6 ft?

  11. Word Problem Practice #3 The height of a javelin (in feet) is modeled by the equation where t is the time in seconds after the javelin is thrown. How long is the javelin in the air?

  12. Drill Solve the following equations.

  13. Drill Simplify each expression.

  14. Drill a. Find the discriminant. b. Identify the number and types of roots. c. Find the solution(s).

  15. The discriminant and solutions*

  16. Drill a. Find the discriminant. b. Identify the number and types of roots. c. Find the solution(s). 1. Simplify. 2. 3.

  17. Divide Polynomials By Binomials (Long Division) Steps for dividing polynomials using long division: Step 1: Divide the first term of the dividend, by the first term of the divisor. Step 2: Multiply the divisor by the quotient and subtract the product from the divisor. Step 3: Bring down the next term and repeat.

  18. Example: Polynomial DivisionNo Remainder

  19. Drill

  20. Example: Polynomial Divisionw/ Remainder

  21. Example: Polynomial DivisionMissing Term

  22. Synthetic Division To divide: 4 2 -13 26 -24 2

  23. Drill

  24. Examples: Synthetic DivisionCoefficient Other than 1

  25. Drill Simplify.

  26. Find (all) the roots • Find the first root by graphing. • Divide by x – root (from part 1) • Find the remaining roots using the quadratic formula.

  27. Drill Activity Sheet 4, #4 a, b, c, and d

  28. The Fundamental Theorem of Algebra • The fundamental theorem of algebra states that every non-constant single-variable polynomial with complexcoefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. • The theorem is also stated as follows: every non-zero, single-variable, degreen polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division.

  29. Drill 1. Find the solutions to the equation, given that one of the roots is 3 . 2. Given , find the remainder of , then find f(2). 3. Given , find the remainder of , then find f(-3).

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