Binomial Theorem Honor’s Algebra II

In the binomial expansion theorem, the numbers, variables, & exponents follow a pattern! (𝒂+𝒃) 𝒏 *Number of terms in your answer = n + 1 *Coefficients follow Pascal's Triangle of Coefficients *Each expansion will begin with "a" and end with "b" & in between them will be "sets" of "ab's" *Exponents will decrease for "a" from left to right and decrease for "b" from right to left *Exponents on each terms add up to the n value

How to set up the binomial expansion theorem: (𝒂+𝒃) 𝒏 Step 1: Write out Pascal’s Triangle of Coefficients in a column. Step 2: Using the exponent, start with a to the “n” power and count down until a is to the 0 power. Step 3: Using the exponent, start with b to the “0” power and count up until b is raised to the “n” power. Step 4: Simplify your terms

Expand the Following 𝑥+2 3

𝑥−4 4 Expand the Following 𝑥−4 4 ***When there is subtraction in the binomial, the signs should alternate!!!

Expand the Following 2𝑥−3 4

Expand the Following 𝑥 2 +4 3

ExPAND THE FOLLOWING: 𝒙−𝟐𝒚 𝟓

Dividing Polynomials

Long division always works! 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 ÷𝐷𝑖𝑣𝑖𝑠𝑜𝑟=𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡 Quotient Divisor Dividend

Setting up for Long Division: 1.) Write your dividend & divisor in STANDARD FORM. *If you are missing an exponent put in a PLACE HOLDER of 0 𝑥 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡 . 2.) On the side of your paper, take the FIRST TERM of the DIVIDEND and divide it by the FIRST TERM of the DIVISOR. Place the quotient ABOVE its LIKE TERM. 3.) Multiply the newest term in the QUOTIENT by each term in the DIVISOR. Place the product "inside" UNDER its LIKE TERM. 4.) Put ( ) around the product and then SUBTRACT. 5.) Bring down the next term in the DIVIDEND & repeat Steps 2 - 5 until the degree of the divisor is bigger than the degree of the dividend. This piece will become your REMAINDER.

What will my answer look like? 𝑄𝑢𝑜𝑡𝑖𝑒𝑛𝑡+ 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 𝐷𝑖𝑣𝑖𝑠𝑜𝑟 Amount Left Over Everything on top of the bar

Example: Divide using long division (2 𝑥 2 −17𝑥−38)÷(2𝑥+3)

Example: Divide using long division (10 𝑥 4 −7 𝑥 2 −1)÷( 𝑥 2 −𝑥+3)

Example: Divide using long division (6 𝑥 3 +13 𝑥 2 −17𝑥+4)÷(3𝑥−1)

How do I Use Synthetic Division??? To divide using Synthetic Division the divisor must be a LINEAR BINOMIAL with a LEADING COEFFICIENT OF 1. (X + #) or (X - #)

What does synthetic division look like?

What does synthetic division look like?

What does my answer look like? The degree of your quotient will always be ONE LESS than the degree of your dividend.

Example: Divide using Synthetic Division ( 𝑥 3 +6 𝑥 2 −30𝑥+120)÷(𝑥+10)

Example : Divide Using Synthetic Division ( 𝑥 3 −20)÷(𝑥−3)

Example : Divide Using Synthetic Division ( 𝑥 2 +5𝑥−14)÷(𝑥−2)

Example : Divide Using Synthetic Division ( 𝑥 3 +𝑥+30)÷(𝑥+3)