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Long Division
Remainder Theorem
Synthetic Division
Rational Zero Theorem
Upper and Lower Bound
100
When dividing a polynomial by a divisor with a power greater than 1.
What is long division?
100
When you divide a polynomial f(x) by x - n the remainder r will be f(n)
The Remainder Theorem
100
You can do this when the divisor is x - n
What is synthetic division?
100
It is: If P(x) is a polynomial with integer coefficients and if P/Q is a zero of P(x) ( P(P/Q) = 0 ), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .
What is the Rational zero Theorem?
100
It is: If you divide a polynomial function f(x) by (x - n), where n > 0, using synthetic division and this yields all positive numbers, then n is an upper bound to the real roots of the equation f(x) = 0.
What is and Upper bound?
200
987 / 7 =
What is 141?
200
It is: Polynomial f(x) divided by x - n = f(x) = (x - n)·q(x) + r(x) The remainder r(x) is always a constant, so therefore... f(x) = (x - n)·q(x) + r When you set x to n... f(n) = (n - n)·q(n) + r Then simplify... f(n) = (0)·q(n) + r
What is the proof of the Remainder Theorem.
200
x + x + 5 / x + 2 is this for synthetic division?
What is not suitable?
200
It is to arrange the polynomial in descending order.
What is first step in finding the zeros?
200
It is: If you divide a polynomial function f(x) by (x - n), where n < 0, using synthetic division and this yields alternating signs, then n is a lower bound to the real roots of the equation f(x) = 0. Special note that zeros can be either positive or negative.
What is the Lower Bound?
300
400 / 8 =
What is 50?
300
It is: When f(n)=0 then x - n is a factor of the polynomial When x - n is a factor of the polynomial then f(n)=0
What is Factor Theorem?
300
x + x + 5 / x^2 + 2 is this for synthetic division.
What is not suitable?
300
It is to write down all the factors of the constant term.
What is 2nd step in finding the zeros?
300
They are: One is n > 0 or positive. The other is that all the coefficients of the quotient as well as the remainder are positive.
What is the two things must occur for n to be an upper bound.
400
It is: f(x) = d(x) ⋅ q(x) + r(x), r(x) = 0 or the degree of r is less than the degree of d.
The Division Algorithm
400
The symbol for this is R.
What is the remainder?
400
To do this, you take all of the coefficients and place them above a line, the divisor's constant to the far left of them.
What is how you set up synthetic division?
400
It is to write down all the factors of the leading coefficient.
What is 3rd step in finding all the zeros?
400
It is: One is n < 0 or negative. The other is that successive coefficients of the quotient and the remainder have alternating signs.
What is the two things must occur for n to be a lower bound?
500
The original problem was: (4x^4 + 3x^3 + 2x + 1)/(x^2 + x + 2)
What was the original problem to this polynomial? 4x^2 - x - 7 R 11x + 15?
500
When a polynomial is divided, it will not have a remainder if the quotient is this.
What is a factor?
500
You place this as a holder when setting up synthetic division, and the polynomial that is ordered from greatest to least power goes from a term with a power that is more than the next term's power by over 1.
What is zero?
500
It is to use synthetic division to determine the values.
What is last step in finding all the zeros?
500
The sin function is this.
What is bounded?