# Synthetica Look And Feel Full 19 EXCLUSIVE

Synthetic Division Synthetic division is another way to divide a polynomial by the binomial x - c , where c is a constant. Step 1: Set up the synthetic division. An easy way to do this is to first set it up as if you are doing long division and then set up your synthetic division. If you need a review on setting up a long division problem, feel free to go to Tutorial 36: Long Division. The divisor (what you are dividing by) goes on the outside of the box. The dividend (what you are dividing into) goes on the inside of the box. When you write out the dividend make sure that you write it in descending powers and you insert 0's for any missing terms. For example, if you had the problem , the polynomial , starts out with degree 4, then the next highest degree is 1. It is missing degrees 3 and 2. So if we were to put it inside a division box we would write it like this: . This will allow you to line up like terms when you go through the problem. When you set this up using synthetic division write c for the divisor x - c. Then write the coefficients of the dividend to the right, across the top. Include any 0's that were inserted in for missing terms. Step 2: Bring down the leading coefficient to the bottom row. Step 3: Multiply c by the value just written on the bottom row. Place this value right beneath the next coefficient in the dividend: Step 4: Add the column created in step 3. Write the sum in the bottom row: Step 5: Repeat until done. Step 6: Write out the answer. The numbers in the last row make up your coefficients of the quotient as well as the remainder. The final value on the right is the remainder. Working right to left, the next number is your constant, the next is the coefficient for x, the next is the coefficient for x squared, etc... The degree of the quotient is one less than the degree of the dividend. For example, if the degree of the dividend is 4, then the degree of the quotient is 3. Example 1: Divide using synthetic division: . Step 1: Set up the synthetic division. Long division would look like this: Synthetic division would look like this: Step 2: Bring down the leading coefficient to the bottom row. *Bring down the 2 Step 3: Multiply c by the value just written on the bottom row. *(-1)(2) = -2 *Place -2 in next column Step 4: Add the column created in step 3. *-3 + (-2) = -5 Step 5: Repeat until done. Step 6: Write out the answer. The numbers in the last row make up your coefficients of the quotient as well as the remainder. The final value on the right is the remainder. Working right to left, the next number is your constant, the next is the coefficient for x, the next is the coefficient for x squared, etc... Example 2: Divide using synthetic division: Step 1: Set up the synthetic division. Long division would look like this: Synthetic division would look like this: Step 2: Bring down the leading coefficient to the bottom row. *Bring down the 1 Step 3: Multiply c by the value just written on the bottom row. *(1)(1) =1 *Place 1 in next column Step 4: Add the column created in step 3. *0 + 1 = 1 Step 5: Repeat until done. Step 6: Write out the answer. The numbers in the last row make up your coefficients of the quotient as well as the remainder. The final value on the right is the remainder. Working right to left, the next number is your constant, the next is the coefficient for x, the next is the coefficient for x squared, etc... Remainder Theorem If the polynomial f(x) is divided by x - c, then the reminder is f(c). This means that we can apply synthetic division and the last number on the right, which is the remainder, will tell us what the functional value of c is. Example 3: Given , use the Remainder Theorem to find f(-2). The steps to the synthetic division are the same as described above. What is different is what are final answer is going to be. This time, we are looking for the functional value, so our answer will not be a quotient, but only the reminder. Using synthetic division to find the remainder we get: Again, our answer this time is not a quotient, but the remainder. Final answer: f(-2) = -27 Factor Theorem If f(x) is a polynomial AND 1) f(c) = 0, then x - c is a factor of f(x). 2) x - c is a factor of f(x), then f(c) = 0. Keep in mind that the division algorithm is dividend = divisor(quotient)+ reminder So if the reminder is zero, you can use this to help you factor a polynomial. If x - c is a factor, you can rewrite the original polynomial as (x - c) (quotient). You can use synthetic division to help you with this type of problem. The Remainder Theorem states that f(c) = the remainder. So if the remainder comes out to be 0 when you apply synthetic division, then x - c is a factor of f(x). Example 4: Use synthetic division to divide by x - 2. Use the result to find all zeros of f. The steps to the synthetic division are the same as described above. What is different is what are final answer is going to be. This time, we are looking for all of the zeros of f. We will start by dividing using synthetic division and then rewrite f(x) as (x - 2)(quotient). Using synthetic division to find the quotient we get: Note how the remainder is 0. This means that (x - 2) is a factor of . Rewriting f(x) as (x - 2)(quotient) we get: We need to finish this problem by setting this equal to zero and solving it: *Factor the trinomial *Set 1st factor = 0 *Set 2nd factor = 0 *Set 3rd factor = 0 The zeros of this function are x = 2, -3, and -1. Example 5: Solve the equation given that 3/2 is a zero (or root) of . The steps to the synthetic division are the same as described above. What is different is what are final answer is going to be. This time, we are looking for all of the zeros of f. We will start by dividing using synthetic division and then rewrite f(x) as (x - 3/2)(quotient). Using synthetic division to find the quotient we get: Note how the remainder is 0. This means that (x - 3/2) is a factor of . Rewriting f(x) as (x - 3/2)(quotient) we get: We need to finish this problem by setting this equal to zero and solving it: *Factor the difference of squares *Note the 1st factor is 2, which is a constant, which can never = 0 *Set 2nd factor = 0 *Set 3rd factor = 0 *Set 4th factor = 0 The solution or zeros of this function are x = 3/2, -1, and 1. Practice Problems

## Synthetica Look And Feel Full 19

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Because of the long time wrist pain ( looking back the starter was the switch to W97 pro staff 315g + stiff LXN ALU PWR 1.25 despite very good feel by play, my wrists and elbows didnt like it after several months ? ? ) I had to switch to a lighter/compromise version of the W97 pro staff L CV ( unstrung weight 290g, strung about 306g ). Now there is the goal for me to find the right stringing to minimize the wrist stress. With heavier racquet the hand/wrist goes by hitting the ball more down that stresses the joints and tendons leading to inflammation, etc.. As a part of my rehab I try to play with the funny Stage3 balls for kids :-), that are lighter a very soft, so the pain especially by playing forehand is tolerable :-).