Section P.3 Operations with Polynomials We will cover: Subtraction Multiplication We will not cover Addition of polynomials or combining like terms. You should know this material from previous algebra courses. Please see page 29, Example 2 part a.

Subtraction Drop the parentheses and distribute the subtraction sign to all of the terms after the subtraction sign. Do these steps at the same time. Note: All terms following the subtraction sign will change to the opposite sign. See next slide for an Example.

Example of Polynomial Subtraction 7 𝑥 4 − 𝑥 2 −4𝑥+2 −(3 𝑥 4 −4 𝑥 2 +3𝑥) 7 𝑥 4 − 𝑥 2 −4𝑥+2−3 𝑥 4 +4 𝑥 2 −3𝑥 (signs in red are now opposite because the subtraction sign has been distributed) Next, reorder terms so that like terms are together 7 𝑥 4 −3 𝑥 4 − 𝑥 2 +4 𝑥 2 −4𝑥−3𝑥+2 Final Answer: 4 𝑥 4 +3 𝑥 2 −7𝑥+2

Another Example of Subtraction − 5 𝑥 2 −1 − −3 𝑥 2 +5 We now have two subtraction signs to distribute: −5 𝑥 2 +1+3 𝑥 2 −5 Let’s reorder the terms: −5 𝑥 2 +3 𝑥 2 +1−5 Final Answer: −2 𝑥 2 −4

A 3rd Example of Subtraction 𝑦 3 +1 −[ 𝑦 2 +1 + 3𝑦−7 ] Let’s add the two polynomials in the brackets first (result shown in blue): 𝑦 3 +1 − 𝑦 2 +3𝑦−6 Next, drop the parentheses and distribute the subtraction sign: 𝑦 3 +1− 𝑦 2 −3𝑦+6 Let’s reorder the terms: 𝑦 3 − 𝑦 2 −3𝑦+1+6 Final Answer: 𝑦 3 − 𝑦 2 −3𝑦+7

A 4th Example of Subtraction 3 𝑥 3 −2 𝑥 2 −5𝑥+2 −4( 𝑥 3 + 𝑥 2 −4𝑥) Distribute the 3 and the 4 first. Then, drop parentheses and distribute the subtraction sign. 3 𝑥 3 −6 𝑥 2 −15𝑥+6 −(4 𝑥 3 +4 𝑥 2 −16𝑥) 3 𝑥 3 −6 𝑥 2 −15𝑥+6−4 𝑥 3 −4 𝑥 2 +16𝑥 Reorder terms so that like terms are together: 3 𝑥 3 −4 𝑥 3 −6 𝑥 2 −4 𝑥 2 −15𝑥+16𝑥+6 Final Answer: − 𝑥 3 −10 𝑥 2 +𝑥+6

Multiplication: Distributing a monomial Example: 𝑦 2 4 𝑦 2 +2𝑦−3 Multiply the 𝑦 2 times each term in the parentheses. I’ve highlighted in red the signs that are carried over from the original problem. (𝑦 2 ) 4 𝑦 2 + (𝑦 2 ) 2𝑦 − (𝑦 2 )(3) Final Answer: 4 𝑦 4 +2 𝑦 3 −3 𝑦 2

Another Example of Distributing a Monomial −3𝑥 𝑥 2 −2𝑥+1 Multiply −3𝑥 times each term inside the parentheses. I’ve highlighted in red the signs that are carried over from the original problem. −3𝑥 𝑥 2 − −3𝑥 2𝑥 +(−3𝑥)(1) −3 𝑥 3 − −6 𝑥 2 + −3𝑥 Final Answer: −3 𝑥 3 +6 𝑥 2 −3𝑥

Multiplying a binomial times a binomial: The FOIL method Multiply using FOIL: (2𝑥−4)(𝑥+5) First terms + Outer terms + Inner terms + Last terms 2𝑥 𝑥 + 2𝑥 5 + −4 𝑥 +(−4)(5) 2 𝑥 2 +10𝑥+ −4𝑥 + −20 2 𝑥 2 +10𝑥−4𝑥−20 Final Answer: 2 𝑥 2 +6𝑥−20

Another Example of FOIL 7𝑥−2 4𝑥−3 First terms + Outer terms + Inner terms + Last terms 7𝑥 4𝑥 + 7𝑥 −3 + −2 4𝑥 +(−2)(−3) 28 𝑥 2 + −21𝑥 + −8𝑥 +6 28 𝑥 2 −21𝑥−8𝑥+6 Final Answer: 28 𝑥 2 −29𝑥+6

Multiplying the sum and difference of the same terms General Rule: 𝑎+𝑏 𝑎−𝑏 = 𝑎 2 − 𝑏 2 Example: 𝑥+5 𝑥−5 FOIL: 𝑥 𝑥 + 𝑥 −5 + 5 𝑥 +5 −5 𝑥 2 −5𝑥+5𝑥−25 (middle terms cancel out) Final Answer: 𝑥 2 −25 Note: 𝑥 2 −25= 𝑥 2 − 5 2 (Recall the General Rule: 𝑎 2 − 𝑏 2 )

Another Example 5𝑥−9 5𝑥+9 FOIL: 5𝑥 5𝑥 + 5𝑥 9 + −9 5𝑥 +(−9)(9) 25 𝑥 2 +45𝑥−45𝑥−81 Final Answer: 25 𝑥 2 −81 Note: 25 𝑥 2 −81= 5𝑥 2 − 9 2 (Recall the General Rule: 𝑎 2 − 𝑏 2 )

The Square of a Binomial Every time you square a two term expression, such as 𝑎+𝑏 2 or a−b 2 , be sure to avoid these common mistakes shown below. Examples of common mistakes: 𝑥+2 2 = 𝑥 2 + 2 2 = 𝑥 2 +4 FALSE 2𝑥−3 2 = 2𝑥 2 − 3 2 =4 𝑥 2 −9 FALSE

Squaring a Binomial is in Fact a FOIL problem: You should always have a middle term when squaring a binomial Examples: 𝑥+3 2 = 𝑥+3 𝑥+3 = 𝑥 2 +3𝑥+3𝑥+9 = 𝑥 2 +6𝑥+9 3𝑥−4 2 = 3𝑥−4 3𝑥−4 =9 𝑥 2 −12𝑥−12𝑥+16 =9 𝑥 2 −24𝑥+16 Note: The last term is ALWAYS positive when you square a binomial. (see signs highlighted in blue)

Shortcut Formulas for squaring a binomial 1st Formula: 𝑎+𝑏 2 = 𝑎 2 +2𝑎𝑏+ 𝑏 2 Example: 2𝑥+1 2 = 2𝑥 2 +2 2𝑥 1 + 1 2 =4 𝑥 2 +4𝑥+1 In previous example, 𝑎=2𝑥 and 𝑏=1. 2nd Formula: 𝑎−𝑏 2 = 𝑎 2 −2𝑎𝑏+ 𝑏 2 Example: 4𝑥−5 2 = 4𝑥 2 −2 4𝑥 5 + 5 2 =16 𝑥 2 −40𝑥+25 In previous example, 𝑎=4𝑥 and 𝑏=5.

Multiplying polynomials where each polynomial has two or more terms Example: (𝑥+2)( 𝑥 2 −3𝑥+4) We will use a technique of setting up a table to perform the multiplication. The first polynomial’s terms are arranged vertically on the left side while the three terms of the second polynomial are arranged along the top of the table. I will insert the signs in the table as well. The terms in each row and column will be multiplied and the result is placed in the appropriate box in red. Note that like terms are arranged diagonally. Answer: 𝑥 3 +2 𝑥 2 −3 𝑥 2 −6𝑥+4𝑥+8= 𝑥 3 − 𝑥 2 −2𝑥+8

Another Example of Table Multiplication Multiply: − 𝑥 2 +𝑥−5 3 𝑥 2 +4𝑥+1 Answer: −3 𝑥 4 +3 𝑥 3 −4 𝑥 3 −15 𝑥 2 +4 𝑥 2 − 𝑥 2 −20𝑥+𝑥−5= −3 𝑥 4 − 𝑥 3 −12 𝑥 2 −19𝑥−5

Cubing a binomial Example of cubing a binomial: 2𝑥+1 3 Common Mistake to Avoid: 2𝑥+1 3 = 2𝑥 3 + 1 3 =8 𝑥 3 +1 FALSE Correct Way: 2𝑥+1 3 = 2𝑥+1 2𝑥+1 2𝑥+1 FOIL the first two: (4 𝑥 2 +4𝑥+1)(2𝑥+1) Finish by using table multiplication. Answer: 8 𝑥 3 +12 𝑥 2 +6𝑥+1