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7.4 Remainder and Factor Theorems 7.5 Roots and Zeros

7.4 Remainder and Factor Theorems 7.5 Roots and Zeros. Algebra II w/ trig. 2 Methods for Polynomial Division can be used to find a quotient and remainder: Long division: will work for divisors of any degree Synthetic Division: is quicker, but only will work for divisors of the form x+k

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7.4 Remainder and Factor Theorems 7.5 Roots and Zeros

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  1. 7.4 Remainder and Factor Theorems7.5 Roots and Zeros Algebra II w/ trig

  2. 2 Methods for Polynomial Division can be used to find a quotient and remainder: • Long division: will work for divisors of any degree • Synthetic Division: is quicker, but only will work for divisors of the form x+k • Long Division: synthetic division:

  3. Find f(3): • Or you can use Synthetic Substitution: If f(x) = -16t2 + 74t + 5

  4. I. REMAINDER THEOREM: If a polynomial f(x) is divided by (x-c), the remainder is f(c). A. Using synthetic substitution(use when degree is greater than 2) to find f(-3) : • if • if • if

  5. II. FACTOR THEOREM: A polynomial f(x) has a factor (x-k) if and only if f(k)=0, so if the remainder is zero. A. Show that (x+5) is a factor of . Then find the remaining factor(s) of the polynomial.

  6. B. Given a polynomial and one of its factors, find the remaining factors of the polynomials. 1.

  7. 2.

  8. 3.

  9. 4.

  10. 7.5 Roots and Zeros • FUNDAMENTAL THEOREM OF ALGEBRA: If f(x) is a polynomial with positive degree, then f(x) has at least one root. • In general: Degree = # of solutions, roots, zeros (but sometimes the same solution can happen more than one (double root - (x+2)2 ; x = -2) Imaginary solutions always occur in pairs: If (a+bi) is a solution, then automatically we have (a – bi) is a solution as well.

  11. I. Given a function and one of its zeros, find the remaining zeros of the functions. A.

  12. B.

  13. C.

  14. D.

  15. E.

  16. II. Write a polynomial equation with the given roots. A. 6, 2i B. 1, 1+i

  17. C. -2, 2+3i D.

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