Week 12 Lesson 1 Polynomials Synthetic Division, Remainder Theorem, Factor Theorem Algebra 2 Honors 10th grade

1. Recognize that you are not dividing by x-a...but you can 2. Factor out the coefficient (which means you are dividing all the terms by the coefficient so the whole number term will become a fraction) on the linear term ("x" term) 3. Divide by "x-a" and then divide by the coefficient you just factored out

What happens if the dividend is not in the form x-a?

It's a factor

What if there's no remainder?

Synthetic division

What is easier long division or synthetic division?

This is when you plug in the number from P(a) using synthetic division to find the remainder You use the number that is in the "a" of the P(a) because that gets you the same thing

What is synthetic substitution and when you use it what number of P(a) do you use?

It would be a negative because it is: x-a x+# x-(-#) so in order for x-a to be positive the "a" term when there is a positive number is negative It is the opposite

What would the "a" be if it was x + #?

The "a" would be positive because it is: x-a x-# in order for it to look like x-a the "a" just needs to be # so it is positive It is the opposite of what is shown

What would the "a" be if it was x-#?

Only when the dividend is linear so it is just an x with an exponent of 1

When can you use synthetic division?

A binomial in the form x-a

What form does the dividend need to be in in order for you to use synthetic division?

First you have to make sure in both the polynomials being divided by each other have there exponents in descending order and if they are not you will have to add a place holder You put the polynomial that is being divided by another polynomial in the house You put the polynomial that is dividing by the other polynomial outside of the house You think what of the first term in the polynomial outside the house gets you the first term of the polynomial inside the house Once you know that you write it on the roof of the house Then you multiply the term on the top of the house by the terms in the polynomial outside the house and put the answers under the like terms in the house You draw a line and change the signs because you are subtracting Then you subtract You then bring the next term in the house down and repeat the steps At the end if there is a remainder you write it over the polynomial outside of the house For your answer you rewrite the division problem and write the polynomial on the top of the house while you make sure the exponents are in descending order

How do you divide polynomials when using long division?

1. Make sure that your exponents are in descending order and that you don't need placeholders 2. Make sure the dividend is in the form x-a and linear which allows you to use synthetic 3. Make the shape of a big L 4. Place the coefficients of the numerator at the top and inside the L 5. Put the "a" value outside and on the left of the L, this is also called on the shelf 6. Bring the first coefficient down under the L and times it by the "a" value 7. Put the product of the coefficient times "a" under the next coefficient, add them together and put the sum under the L 8. Repeat the process again for this sum 9. When you get to the last one draw a dashed line between it and the rest to separate the remainder that is the last number at the end on the bottom of the L 10. Write the division problem out with the equal sign 11. Since P(x) is an x^3 and you are dividing it by an x the first term is an x^2 so start with that and write it under the coefficients at the bottom of the L in decreasing order 12. Write the answer with the coefficients and the terms and write the remainder over D(x)

How do you divide two polynomials by synthetic division?

If the remainder is 0

How do you know if the polynomial is a factor?

Write the binomial you divided by next to the polynomial you got from using synthetic division That is the product (they are multiplying by each other) of the factors (the factors of the polynomial)

How do you write the polynomial as a product of factors if it is a factor?

Remainder Theorem

If a polynomial P(x) is divided by x-a, the remainder is P(a) You can find P(a) by using synthetic division which is easier and this is called synthetic substitution

Multiplying by a fraction

In the second step of division when the dividend binomial doesn't = x-a when you divide the polynomial answer of the synthetic division by the coefficient you factored out what is it the same as?

Since the coefficient is a fraction the 1 gets multiplied by the number in the numerator of the remainder and the number in the denominator in the coefficient gets distributed into the binomial in the denominator of the remainder

In the second step of synthetic when you need to change it to =x-a if there is a remainder how do you multiply by the coefficient?

What you divided by which is the x-a term

In the second step of synthetic when you need to change it to =x-a if there is a remainder what do you write it over?

Factor theorem

The expression x-a is a factor of a polynomial P(x) if and only if P(a)= 0 You can find factors by using synthetic division and getting a remainder of 0 or by using P(a)=0

Divide the coefficient by all terms is factoring it out so remember to make the whole number a fraction

When using synthetic division and the binomial dividend is not in x-a form when you factor out the coefficient what do you need to remember to do?

They both have a remainder of 0 which means it's a factor

When you plug P(1) into the polynomial and it is also divided by x-1 with synthetic division what is the same

The remainder

When you plug in P(a) as P(2) into P(x) and use synthetic division with the polynomial with a dividend of x-2 so you put 2 on the shelf what is the same?

Since you are just multiplying by 1 all the coefficients are going to end up being the same so you can just add the coefficients together to get your answer If it was -1 you would change the signs

When you plug in a 1 to a polynomial like this what is a shortcut that you can do?

Don't forget the signs

When you put the coefficients in the L for synthetic division what do you need to make sure to do?