Long & Synthetic Division

1st write as a division problem.

Correct x2 What would you multiply x by to get x3 ?

Multiply x2 by (x-7) and write under dividend Then subtract by adding the opposite

Correct 5x What would you multiply x by to get 5x2 ?

Multiply 5x by (x-7) and write under the previous difference Then subtract by adding the opposite

Correct 40 What would you multiply x by to get 40x?

Multiply 40 by (x-7) and write under the previous difference Then subtract by adding the opposite

This is the remainder

Show x-3 is a factor of then rewrite with x-3 times the quotient Because the remainder is zero x-3 is a factor

Show x-5 is a factor of then rewrite with x-5 times the quotient Because the remainder is zero x-5 is a factor

P343 1 – 13 odd Divide using long division. Write your answer as

Write the coefficients of the numerator adding any place holders Use Synthetic Division to do the long division problem Write the opposite of the constant term 1 7 35 280 5 40 284 Write the coefficients of the numerator adding any place holders Multiply by 7 Add Drop the Coefficient Down Synthetic Division only works when the denominator is x-a where a is a constant

Synthetic Division 7 35 280 1 5 40 284 The quotient will always be degree 1 less than the degree of the numerator

Synthetic Division Write the opposite of the constant term 7 35 280 284 1 5 40 These are the coefficients of the quotient This is the remainder

Use Synthetic Division to divide 3 -12 40 -120 -120 -10 30 30 -20 Multiply by -4 Add

Use synthetic division 2 4 8 14 28 4 7 14 33

Remainder Theorem: Let f be a polynomial function Remainder Theorem: Let f be a polynomial function. If f(x) is divided by x-c, then the remainder is f(c) If you divide a polynomial function synthetically by a number the remainder is the value that would result if you plugged that number in the equation for x These two numbers will always be the same 2 1 -6 0 -7 5 3 1 -4 -8 -23 -41 -79 2 -8 -16 -46 -82

= -305 Use synthetic division to find 3 -2 -7 4 3 1 -2 -13 -35 -102 -305 -6 -39 -105 -306

Factor Therorem:Let f be a polynomial function Factor Therorem:Let f be a polynomial function. Then x-c is a factor of f(x) if and only if f(c)=0 Since f(5)=0 then synthetically dividing by 5 will give a remainder of 0 and x-5 has to be a factor of f(x) 5 2 -3 -35 10 35 2 7 0

Is x-3 is a factor of : If (x-3) is a factor of g(x) when we divide it into g(x) it will have no remainder. 3 1 -3 -4 12 3 0 -12 1 0 -4 0 So we can write g(x) as What does g(3)=

Show x+1 is a factor of g(x) then write g(x) as the product of x+1 and the quotient -1 4 -4 What does g(-1)=

Write answers in the form P344 21,23,27,29,31,33,35,37