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6.5: The Remainder and Factor Theorems

6.5: The Remainder and Factor Theorems. Objectives: Students will be able to…. Divide polynomials using long division and synthetic division Determine factors of a polynomial using the factor and remainder theorems Find all zeros of a polynomial. . Dividing Polynomials.

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6.5: The Remainder and Factor Theorems

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  1. 6.5: The Remainder and Factor Theorems Objectives: Students will be able to…. Divide polynomials using long division and synthetic division Determine factors of a polynomial using the factor and remainder theorems Find all zeros of a polynomial.

  2. Dividing Polynomials • The expression you divide by is the divisor(d(x)) • Quotient is the resulting polynomial (q(x)) • The degree of the remainder will be less than the divisor • f(x) is the dividend

  3. Long Division: • Make sure you put a “0” as a coefficient if there is a missing degree • Divide first term in the divisor into term with highest degree in what’s left in dividend • You stop the process when the expression left in dividend is less than the degree of the divisor.

  4. Example:

  5. Synthetic Division • Can use when dividing by a polynomial in the form x – k • Use the value of k (opposite sign of what is in the binomial) • Reintroduce the variables in the quotient (will always be 1 degree less than the dividend) • Last number in lower right corner is remainder

  6. Using Synthetic Division

  7. Examples: Use Synthetic Division 1. 2.

  8. The Remainder Theorem If a polynomial f(x) is divided by x – k, then the remainder r= f(k). (Remember we used synthetic substitution to evaluate a function!!)

  9. IMPORTANT: IF THE REMAINDER IS 0, THEN THE DIVISOR IS A FACTOR OF THE DIVIDEND. (it divides evenly into the polynomial, and you use it to find other factors by factoring the quotient)

  10. Factor Theorem A polynomial f(x) has a factor x – k if and only if f(k) = 0. If x-k is a factor (meaning the remainder is 0), then x = k is a zero!!!!

  11. We can use zeros to factor a polynomial completely Find all factors of f(x) = 3x3 + 13x2+2x-8 given that f(-4)=0. • If f(-4)= 0 then x – (-4) is a factor

  12. One zero of f(x) = x3 +6x2 +3x -10 is x = -5. Find the other zeros of the function.

  13. Factor f(x) = 3x3 + 14x2-28x-24 given that f(-6)=0.

  14. One zero of f(x) = 2x3 -9x2 -32x -21 is x = 7. Find the other zeros of the function.

  15. The volume of a rectangular prism is V(x) = 2x3 + 17x2 +40x +25. The length is x + 5 and the height is x+1. What is the width?

  16. Critical Thinking… How would you find all zeros of f(x)=x3+x2+2x+24 knowing that x = -3 is a zero?

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