http://www.ck12.org Chapter 2. Polynomials and Rational Functions
x^3 + 2 xx^2 −− 24 x+ 8- Completely factor the following polynomial.
x^4 + 6 x^3 + 3 x^2 − 26 x− 24 - Divide the following polynomials.
3 x^5 − 2 xx^2 −+ 110 x− 5
Answers:
1.x^3 +^2 xx−^2 − 24 x+^8 =x^2 + 4 x+ 4 +x^16 − 2
- Notice that possible roots are± 1 , 2 , 3 , 4 , 6 , 8 ,24. Of these 14 possibilities, four will yield a remainder of
zero. When you find one, repeat the process.
x^4 + 6 x^3 + 3 x^2 − 26 x− 24
= (x+ 1 )(x^3 + 5 x^2 − 2 x− 4 )
= (x+ 1 )(x− 2 )(x^2 + 7 x+ 12 )
= (x+ 1 )(x− 2 )(x+ 3 )(x+ 4 )3.^3 x^5 −^2 xx−^2 + 110 x−^5 = 3 x^4 + 3 x^3 + 3 x^2 +x+ 11 +x−^61
Practice
Identify all possible rational solutions of the following polynomials using the Rational Root Theorem.
- 15x^14 − 12 x^13 +x^12 +···+ 2 x^2 − 5 x+ 5 = 0
- 18x^11 + 42 x^10 +x^9 +···+x^2 − 3 x+ 7 = 0
- 12x^16 + 11 x^15 + 3 x^14 +···+ 6 x^2 − 2 x+ 11 = 0
- 14x^7 − 7 x^6 +x^5 +···+x^2 + 6 x+ 3 = 0
- 9x^9 − 10 x^8 + 3 x^7 +···+ 4 x^2 − 2 x+ 2 = 0
Completely factor the following polynomials. - 2x^4 −x^3 − 21 x^2 − 26 x− 8
7.x^4 + 7 x^3 + 5 x^2 − 31 x− 30
8.x^4 + 3 x^3 − 8 x^2 − 12 x+ 16 - 4x^4 + 19 x^3 − 48 x^2 − 117 x− 54
- 2x^4 + 17 x^3 − 8 x^2 − 173 x+ 210
Divide the following polynomials.
11.x^4 +^7 x^3 +x^5 +x^24 −^31 x−^30
12.x^4 +^7 x^3 +x^5 +x^22 −^31 x−^30
13.x^4 +^3 x^3 −x^8 +x^23 −^12 x+^16
14.^2 x^4 −x 3 x−^3 −x (^221) −x 102 −x−^268 x−^8
15.x^4 x+ (^3) +^8 x 103 +x^32 x+^2 − 2332 x+x 14 −^28