Polynomial operations Integrated mathematics

Types of Polynomials Name # Terms Example Monomial Binomial Trinomial

Like Terms Same Variable Same Exponent Example: 2𝑥 2 and 3 𝑥 2

Degree: Largest exponent on polynomial 5 𝑥 3 3 𝑥 5 +4 𝑥 2 5 𝑥 3 +2 𝑥 2 +4

Descending Order: Highest degree first 3 𝑥 2 +4 𝑥 5 −7𝑥 5𝑦 −9−2 𝑦 4 −6 𝑦 3

Descending Order: 3 𝑥 4 𝑦 2 +2 𝑥 5 𝑦 2 −8 𝑥 3 𝑦 2 3 𝑥 4 𝑦 2 +2 𝑥 5 𝑦 2 −8 𝑥 3 𝑦 2 3 𝑥 3 𝑦−8𝑥 𝑦 3 +5 𝑥 4 𝑦 4

Try on your own 𝟐 𝒙 𝟑 −𝟓𝒙+𝟕 𝒙 𝟓 𝟒 𝒙 𝟒 𝒚−𝒙+𝟕 𝒙 𝟔 𝟐 𝒙 𝟑 −𝟓𝒙 𝒚 𝟒 +𝟕 𝒙 𝟕 4 𝒂 𝟔 +𝟗𝒙𝒚𝒛+𝟕 𝒙 𝟓

Example 1 3𝑥 2 +2𝑥−2 +( −2𝑥 2 +5𝑥+5)

Example 2 31𝑚 4 + 𝑚 2 +2𝑚−1 +( −7𝑚 4 + 5𝑚 2 −2𝑚+2)

Example 3 4𝑎 2 𝑏−5𝑎+2 +( −2𝑎 2 𝑏−2𝑎−4)

Example 4 3𝑛 3 − 3𝑚 3 𝑛 2 −5𝑛−3 +( 5𝑛 3 + 2𝑚 3 𝑛 2 −3𝑚−2𝑛−2)

Example 5 −2𝑚 3 − 5𝑚 2 −2𝑚−4 +( 𝑚 4 − 6𝑚 2 +7𝑚−10)

Example 6 −2𝑥 4 𝑦 3 −5𝑥𝑦+2 +( 𝑥 4 𝑦 3 + 𝑥 2 +2𝑥𝑦+5)

Try on your own 2𝑥 2 +7𝑥+3 +( −5𝑥 2 −3𝑥−6) 𝟒 𝒙 𝟒 −𝒙+𝟕 +(𝟗 𝒙 𝟒 +𝟑𝒙−𝟕) 𝟔 𝒙 𝟑 −𝟑𝒙 𝒚 𝟒 +𝟕 +(𝟐 𝒙 𝟑 −𝟓𝒙 𝒚 𝟒 +𝟕 𝒙 𝟕 ) 𝟑 𝒙 𝟐 𝒛− 𝟓𝒚 𝟒 𝒙+𝟑𝒙 + −𝟐 𝒙 𝟐 𝒛− 𝒚 𝟒 𝒙−𝟐𝒙

PDN Identify the mistake and correct the problem. 4𝑥 4 +3 𝑥 2 −3𝑥 + 2 𝑥 2 −5𝑥+2 =6 𝑥 2 −2 𝑥 2 −1𝑥

Subtract Example 7 𝑎 3 −2 𝑎 2 +4 −( 𝑎 4 −4 𝑎 3 − 3𝑎 2 )

Example 8 3𝑥 3 𝑦 2 −4𝑥𝑦+1 −(−4 𝑥 3 𝑦 2 −3 𝑥 2 𝑦 2 +3𝑥𝑦−5)

Example 9 4 𝑥 3 +2 𝑥 2 −2𝑥−3 −(2 𝑥 3 −3 𝑥 2 +2)

Ticket out the door 2𝑥 2 +7𝑥+3 −( −5𝑥 2 −3𝑥−6) 𝟑 𝒙 𝟐 𝒛− 𝟓𝒚 𝟒 𝒙+𝟑𝒙 − −𝟐 𝒙 𝟐 𝒛− 𝒚 𝟒 𝒙−𝟐𝒙

PDN 2 𝑥 2 +2𝑥−7 − 2 𝑥 2 −2𝑥−3 2 𝑥 2 +2𝑥−7 + 2 𝑥 2 +2𝑥+3 =4 𝑥 2 +4𝑥−4 Find and describe the mistake. Then, correct. 2 𝑥 2 +2𝑥−7 − 2 𝑥 2 −2𝑥−3 2 𝑥 2 +2𝑥−7 + 2 𝑥 2 +2𝑥+3 =4 𝑥 2 +4𝑥−4

Multiplying Polynomials Simplify using the distributive property Example 1 2x ( 5x + 3 )

Simplify using the distributive property Example 2 − 𝟓𝒙 𝟐 (𝟑𝒙+𝟓)

Simplify using the distributive property Example 3 𝟑𝒂 𝟐 ( −𝟓𝒂 𝟑 +𝟐𝒂−𝟕)

Simplify using the distributive property Example 4 8p( 𝟑𝒒 𝟒 −𝟐 𝒒 𝟑 𝒑 𝟐 +𝟐𝒑)

Try on your own 𝟑 𝒙−𝟕 𝟐𝒙 𝟐 ( 𝟓𝒙 𝟐 −𝟕𝒙+𝟏) −𝟖𝒚(−𝟐𝒚 𝟐 −𝟕)

Ticket out the door 𝟓𝒕(𝟐𝒕−𝟒) 𝟒𝒙𝒚 𝟐 (𝟐 𝒙 𝟓 −𝟒𝒚)

PDN 3 𝑥 2 𝑦 3 𝑥 4 𝑦−4𝑥 =9 𝑥 2+4 𝑦−12 𝑥 2+1 𝑦 =9 𝑥 6 𝑦−12 𝑥 3 𝑦 Find and describe the mistake. Then, correct. 3 𝑥 2 𝑦 3 𝑥 4 𝑦−4𝑥 =9 𝑥 2+4 𝑦−12 𝑥 2+1 𝑦 =9 𝑥 6 𝑦−12 𝑥 3 𝑦

Multiplying Binomials x ( x + 5 ) 1 ( x + 5 )

How would you use the distributive property to simplify this? Ex. 5 ( x + 1) ( x + 5 )

F O I L Ex. 6 ( 2n + 3) ( n - 6 )

Ex. 7 (𝒄−𝟓)(𝟑𝒄+𝟏)

Ex. 8 (𝟐𝒙−𝟏)(𝟑𝒙−𝟏)

Ticket out the door (𝟐𝒙−𝟓)(𝒙+𝟕) (𝒃−𝟏)(𝒃−𝟗)

PDN (3𝑥−2) 𝑥−4 =3𝑥−12𝑥−2𝑥+8 =−11𝑥+8 Find and describe the mistake. Then, correct. (3𝑥−2) 𝑥−4 =3𝑥−12𝑥−2𝑥+8 =−11𝑥+8

SPECIAL CASES 𝐚+𝐛 𝐚 −𝐛 = 𝐚 −𝐛 𝐚+𝐛 = 𝐚 𝟐 − 𝐛 𝟐 𝐚+𝐛 𝐚 −𝐛 = 𝐚 −𝐛 𝐚+𝐛 = 𝐚 𝟐 − 𝐛 𝟐 (𝒂+𝒃 ) 𝟐 = 𝒂 𝟐 + 2ab + 𝒃 𝟐 (𝒂 −𝒃) 𝟐 = 𝒂 𝟐 - 2ab + 𝒃 𝟐

Special Cases Example 9 (𝒚+𝟓)(𝒚−𝟓)

Special Cases Example 10 (𝟒𝒙+𝟕)(𝟒𝒙−𝟕)

Special Cases Example 11 ( 𝒚−𝟓) 𝟐

Special Cases Example 12 ( 𝟐𝒗 𝟐 −𝟖) 𝟐

Try on your own (𝟐𝒙−𝟓) 𝟐 (𝒙−𝟓) 𝟐 (−𝟐𝒙+𝟕) 𝟐 (𝟑𝒙−𝟏) 𝟐

Binomials & Trinomials Example 1 (𝒙+𝟏)( 𝒙 𝟐 +𝒙+𝟏)

Binomials & Trinomials Example 2 ( 𝒙 𝟐 +𝟒)( 𝒙 𝟐 +𝟐𝒙−𝟑)

Binomials & Trinomials Example 3 (𝟑𝒙+𝟐)(𝟑 𝒙 𝟐 +𝟐𝒙−𝟏)

Binomials & Trinomials Example 4 (𝟑𝒄−𝟒)( 𝟐𝒄 𝟐 −𝒄+𝟑)

Try on your own (𝟐𝒕−𝟖)( 𝟑𝒕 𝟐 −𝒕+𝟒) (𝟓𝒗+𝟐)( 𝟑𝒗 𝟐 +𝟐𝒗−𝟖) ( 𝟒𝒅 𝟐 +𝟑)( 𝟐𝒅 𝟐 +𝒅+𝟓) (𝟑𝒃−𝟕)( 𝟓𝒃 𝟒 −𝟓𝒃−𝟗)