1. 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.

2. 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.

3. 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.

4. 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

5. 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.

6. 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 ?
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
𝟎.𝟎𝟎𝟖= (𝟎.𝟐) 𝟑
−𝟏𝟎𝟎𝟎= (−𝟏𝟎) 𝟑
𝟏 𝟐𝟕 = 𝟏 𝟑 𝟑

7. 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 ? 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
−𝟎.𝟎𝟎𝟎𝟏 is negative.
𝟏𝟔 𝟐𝟕 = 𝟐 𝟑 𝟒
𝟏= (𝟏) 𝟒
There are NO square roots of
−𝟏= (−𝟏) 𝟒
− 𝟏𝟔 𝟐𝟕 = − 𝟐 𝟑 𝟒

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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