Mrs. Rivas Round Table 𝒙 𝟓 −𝟓 𝒙 𝟒 +𝟒 𝒙 𝟑 +𝟐 𝒙 𝟑 =𝟎 Ida S. Baker H.S. Round Table Solving polynomial equations using factors. What are the solutions of 𝑥 5 +4 𝑥 3 =5 𝑥 4 −2 𝑥 3 ? What are their multiplicities? 𝒙 𝟓 −𝟓 𝒙 𝟒 +𝟒 𝒙 𝟑 +𝟐 𝒙 𝟑 =𝟎 𝒙 𝟓 −𝟓 𝒙 𝟒 +𝟔 𝒙 𝟑 =𝟎 𝒙 𝟑 ( 𝒙 𝟐 −𝟓𝒙+𝟔)=𝟎 Factor out the GCF. 𝒙 𝟑 (𝒙−𝟑)(𝒙−𝟐)=𝟎 Factor 𝒙²−𝟓𝒙−𝟔. 𝒙 𝟑 =𝟎 𝒙−𝟑=𝟎 𝒙−𝟐=𝟎 𝒙=𝟎 𝒙=𝟎 𝒙=𝟎 𝒙=𝟑 𝒙=𝟐 The numbers 0 are zeros of multiplicity 3. The numbers 3, and 2 are zeros of multiplicity 1.

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Objective: To find nth roots. Essential Question: What is the nth root of a number? The nth root of a number is a number that when raised to the nth power gives you the original number. Steps: 1. Write the nth root of b as 𝑛 𝑏 . 2. Use the property of nth roots of nth powers to simplify radical expressions.

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Objective: To find nth roots. Corresponding to every power there is a root. Example: 5 is a square root of 25. 𝟓 𝟐 =𝟐𝟓 5 is a cube root of 125. 𝟓 𝟑 =𝟏𝟐𝟓 5 is a fourth root of 625. 𝟓 𝟒 =𝟔𝟐𝟓 5 is a fifth root of 3125. 𝟓 𝟓 =𝟑𝟏𝟐𝟓 This pattern suggests a definition of an nth root.

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. 𝒂 𝒏 =𝒃 𝑰𝒇 𝒏 𝒊𝒔 𝒐𝒅𝒅… 𝑰𝒇 𝒏 𝒊𝒔 𝑬𝒗𝒆𝒏… there is 𝐨𝐧𝐞 𝐫𝐞𝐚𝐥 𝒏𝒕𝒉 root of 𝒃, denoted in radical form as 𝑛 𝑏 . and 𝒃 𝐢𝐬 𝐩𝐨𝐬𝐢𝐭𝐢𝐯𝐞, there are two real nth roots of b. The positive root is the PRINCIPAL ROOT and its symbol is 𝑛 𝑏 . The negative root is its opposite, or − 𝑛 𝑏 . 𝒐𝒅𝒅 𝒃 = 𝟑 𝟐𝟕 =𝟑 𝒆𝒗𝒆𝒏 𝒃 = 𝟐 𝟏𝟔 =𝟒 𝒐𝒅𝒅 −𝒃 = 𝟑 −𝟐𝟕 =−𝟑 and b is negative, there are NO real nth roots of b. The only nth root of 0 is 0. 𝒆𝒗𝒆𝒏 −𝒃 = 𝟐 −𝟏𝟔 =𝑵𝒐 𝒓𝒆𝒂𝒍 𝒓𝒐𝒐𝒕

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Gives the degree of the root. Number under the radical sign.

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Example # 1 Finding all real roots. A What are the real cube roots of 0.008, −1000, and 1 27 ? 𝟎.𝟎𝟎𝟖 −𝟏𝟎𝟎𝟎 𝟏 𝟐𝟕 𝟎.𝟐 is the only real cube root of 0.008 −𝟏𝟎 is the only real cube root of −1000 𝟏 𝟑 is the only real cube root of 1 27 . 𝟎.𝟎𝟎𝟖= (𝟎.𝟐) 𝟑 −𝟏𝟎𝟎𝟎= (−𝟏𝟎) 𝟑 𝟏 𝟐𝟕 = 𝟏 𝟑 𝟑

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Example # 1 Finding all real roots. B What are the real fourth roots of 1, −0.0001, and 16 81 ? 𝟏 −𝟎.𝟎𝟎𝟎𝟏 𝟏𝟔 𝟖𝟏 For an even n and 𝑏 <0, there are no real nth roots. 𝟏 is a real fourth root of 1. 𝟐 𝟑 is the only real cube root of 16 81 . −𝟎.𝟎𝟎𝟎𝟏 is negative. 𝟏𝟔 𝟐𝟕 = 𝟐 𝟑 𝟒 𝟏= (𝟏) 𝟒 There are NO square roots of 0.0001. −𝟏= (−𝟏) 𝟒 − 𝟏𝟔 𝟐𝟕 = − 𝟐 𝟑 𝟒

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. You Do It Finding all real roots. What are the real fifth roots of 0, −1, and 32? 𝟎 −𝟏 𝟑𝟐 −𝟏 is a real fifth root of 1. 𝟎 is a real fifth root of 1. 𝟐 is the only real fifth root of 32. 𝟎= (𝟎) 𝟓 −𝟏= (−𝟏) 𝟓 𝟑𝟐= 𝟐 𝟓

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Example # 2 Finding roots. What is each real root? A 𝟑 −𝟖 −𝟐 𝟐 B 𝟎.𝟎𝟒 C 𝟒 −𝟏 D (−𝟐) 𝟑 =−𝟖 (𝟎.𝟐) 𝟐 =𝟎.𝟎𝟒 There are NO real root because is no real whole fourth power −𝟏. −𝟐 𝟐 = 𝟒 So, 𝟑 −𝟖 =−𝟐 So, 𝟎.𝟎𝟒 =𝟎.𝟐 =𝟐

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. You Do It Finding roots. What is each real root? A 𝟑 −𝟐𝟕 −𝟒𝟗 B 𝟒 −𝟖𝟏 C −𝟕 𝟐 D (−𝟑) 𝟑 =−𝟐𝟕 There are NO real root because is no real whole fourth power −𝟖𝟏. −𝟕 𝟐 = 𝟒𝟗 There are NO real root because is no real whole fourth power −𝟒𝟗. So, 𝟑 −𝟐𝟕 =−𝟑 =𝟕

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. 𝒏 𝒂 𝒏 𝒏 𝒂 𝒏 Means: You must Include the absolute value 𝒂 when n is EVEN. 𝒙 𝟒 𝒚 𝟔 = 𝒙 𝟐 𝒚 𝟑 𝟐 = 𝒙 𝟐 𝒚 𝟑 = 𝒙 𝟐 𝒚 𝟑 You must Omit the absolute value when n is ODD. 𝟑 𝒙 𝟑 𝒚 𝟔 = 𝟑 𝒙 𝒚 𝟐 𝟑 =𝒙 𝒚 𝟐

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Example # 1 Simplifying radical expressions. What is a simpler form of each radical expression? A 𝟏𝟔 𝒙 𝟖 The index 2 is even, USE absolute value symbols 𝟏𝟔 𝒙 𝟖 = 𝟒 2 𝒙 𝟒 2 |4 𝑥 4 | = 4 𝑥 4 because 𝑥 4 is never negative. = 𝟒𝒙 𝟒 2 = 𝟒𝒙 𝟒 =𝟒 𝒙 𝟒

The index 3 is ODD, DO NOT USE absolute value symbols Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Example # 1 Simplifying radical expressions. What is a simpler form of each radical expression? B 𝟑 𝒂 𝟔 𝒃 𝟗 The index 3 is ODD, DO NOT USE absolute value symbols 𝟑 𝒂 𝟔 𝒃 𝟗 = 𝟑 𝒂 𝟔 𝒃 𝟗 𝟑 = 𝟑 𝒂 𝟐 𝟑 𝒃 𝟑 𝟑 = 𝒂 𝟐 𝒃 𝟑

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. Example # 1 Simplifying radical expressions. What is a simpler form of each radical expression? C 𝟒 𝒙 𝟖 𝒚 𝟏𝟐 The index 4 is Even, USE absolute value symbols 𝟒 𝒙 𝟖 𝒚 𝟏𝟐 = 𝟒 𝒙 𝟐 𝟒 𝒚 𝟑 𝟒 | 𝑥 2 | = 𝑥 2 because 𝑥 2 is never negative. = 𝟒 𝒙 𝟐 𝒚 𝟑 𝟒 = 𝒙 𝟐 𝒚 𝟑 = 𝒙 𝟐 𝒚 𝟑

Roots and Radical Expressions. Section 6-1 Roots and Radical Expressions. Mrs. Rivas Ida S. Baker H.S. You Do It Simplifying radical expressions. What is a simpler form of each radical expression? C 𝟒 𝒙 𝟏𝟐 𝒚 𝟏𝟔 A 𝟖𝟏 𝒙 𝟒 B 𝟑 𝒂 𝟏𝟐 𝒃 𝟏𝟓 = 𝟗𝒙 𝟐 = 𝒂 𝟒 𝒃 𝟓 = 𝒙 𝟑 𝒚 𝟒