Division and Rational Root Theorem TS Making decisions

Division and Rational Root Theorem TS: Making decisions after reflection and review Obj: Review polynomial division and how to find the roots of polynomials Warm-Up: Think back to Algebra 2, what we have done so far and what you know about zeros and points on a function and answer the following. Find the cubic equation with zeros of -2, 1 and 5 and passes through the point (-1, 24)

Two kinds of division: Long Division – When is it used? Synthetic Division – When is it used?

Synthetic Division vs. Long Division 1) (x 7 + 1) ÷ (x + 1)

Synthetic Division vs. Long Division 2)(5 x 3 + 6 x + 8) ÷ (x + 2)

Synthetic Division vs. Long Division 3) (4 x 4 + 8 x) ÷ (x 2 – 1)

The Remainder Theorem If a polynomial f(x)is divided by x – k, the remainer is r = f(k). Rational Root Theorem If the polynomial f(x) = anxn + an-1 xn-1 + … + a 2 x 2 + a 1 x + a 0 Has integer coefficients, every rational zero of f has the form Rational zero = Where p and q have no common factors other than 1, p is a factor of the constant term a 0 and q is a factor of the leading coefficient an

Given P(x) = x 6 – 2 x 4 + x 3 – 5 x 2 + 1 Find P(2)

Given f(x) = x 3 – 2 x 2 +a x – 4 If f(2) = 4, what is a?

Factor x 3 + 3 x 2 – 10 x – 24

Find all the zeros of x 5 – x 4 – 3 x 3 + 5 x 2 - 2 x = 0

Now You Try: 1) Fully factor 8 x 4 – 14 x 3 – 71 x 2 – 10 x + 24 (How do I check my solution with my calc I wonder? ? ? ) 2) What is the remainder of (x-3)(x-4)(x-1)4 divided by x? Ans: 12