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] DeMA [DEPARTAMENT DE MATEMÀTIQUES IES Clot del Moro
UD1 – ALGEBRA, OPERATIONS WITH POLYNOMIALS.
EXERCISES TO PRACTICE.
1. Indicate the coefficient, literal part, exponent and degree of each monomial:
a) 3 x^2 b) − 2 x^7 c) x d) 5 e)
x 2
2. Which of the following expressions are monomials? Indicate the degree in each case.
a) 5 x
3
y b) 3 x
2 y 3
c) 7 x
2
d) 4
e)
x
2 f)^
x 2
g) 3 + x h) x·y^2
3. Indicate which of the following monomials are similar to 5 x^2 :
7 x 2 5 x (^3 5) x 5 xy x
x 2 − 9 x 2
4. Write two monomials similar to each of the following:
a) − 5 x^2 b) 2 x^4 c) x d) 3 x^2 y
5. Find the numerical value of the following polynomials when x = 3 e y =− 2 :
a) 5 x
3
b) 2 y^ c) 2 xy^ d) − xy
2
6. Reduce (add and/or subtract) the following monomials when possible:
a) 5 x − 3 x + 4 x + 7 x − 11 x + x b) 7 x^3 − 3 x^2 + 4 x^2 + 7 x^2 − 11 x^3 + x^3
c) 5 x^2 − 3 x + 4 + 7 x^2 − 11 x − 6 c) 3 x^2 y − 5 x^2 y + 2 x^2 y + x^2 y
7. Operate:
a) ( 3 x
2 ) · ( 5 x 4
) d) ( x
2
) · ( x ) g) ( 5 x
3 ) 2
b) ( 2 x )^4 e) ( 5 x^3 ) :( x^2 ) h) ( 15 x^6 ):( 5 x^2 )
c) ( 12 x
3 y 2
):( 3 x ) f) ( 15 x
4 y 7 ):( 3 xy 5
) i) ( 2 x
3 y 2 ) 2 :( 2 x 4 y 3 )
8. Divide:
a)
6 x ⁴
2 x
b)
8 x⁵
− 2 x
c)
−10x²
− 2 x
d)
− 56 x⁶
− 7 x
e)
63 a ⁴ b
3 a
f)
− 16 x⁹
4 x²
g)
33 a⁵b ⁷
− 3 a² b ⁴
h)
125 z⁸
5 z⁶
i)
60 a⁶ b⁶
5 a² b
j)
− 49 r³ s⁵ t⁴
7 r² s² t²
k)
6 x³ y ⁴ z²
3 x² y² z²
l)
36 x³ y z²
12 x ⁵ y² z
9. Simplify the following expressions by applying the operations with monomials:
a) x^ ^ x
b) y^ ^ y^ ^ y
c) 5 y^ ^2 y
d) 2 x^^2 ^11 x^^2 ^3 x^2^^ ^4 x^2
e) 3 x^ ^4 x =
f) x^3^ x^2 x^5^ x^7 =
g) ^4 a bc^3^^2 ^2 a^5 =
h) x^ ^ (^ x^2 )^3 =
i)
(2 a^2 ) 3 (3 ) x^2
j) 5 a b^2^ ^3 a b^2 =
k) 3 a^2 2 a =
l) 4 x^4 ⋅ 6 x^7 =
m) 4 x^3 ⋅(− 12 ) x^5 =
n)
x^5 ⋅
x^7 =
o) 5(^ c^ ^ 4) =
p) 10( 9^ ^ ^ 4 ) x =
q) ^ z^ ^ z^ ^ z^ ^ z
r)^ b^2^^ ^ b^^2 ^ b^2^^ ^ b^^2 ^ b^2
s) 2 c^ ^3 c^ ( 4 )^ c
t) 2 a 3 b 1 6 a 5 b 4
u) 3 (^ x ) ( 5)^ ^ ^ x x ^
v)
4 x^3^ 3 x^2 ( 5 ) x
w)
8 x ( 4 x^2 )( 2 ) x
x) abc bca cab
10. Reduce:
14. Consider P ( x )= 3 x
2
− 2 x + 3 and Q (^ x )=^2 x −^3 to calculate:
a) P ( 2 )=
b) P ( 3 )− Q ( 4 )=
c) 2 · P (− 3 )− 3 ·Q (− 1 )=
d) P (− 1 )=
e) Q ( 5 )− P (− 2 )=
f) 5 · P ( 1 )+ 2 ·Q (− 2 )=
15. Add the following polynomial expressions:
a) (3 x^2 11 x 4) ( x^2 3 x 2)
b) (3 x^4^ 7 x 1) (4 x^4 3 x^2 2 x 3)
c) (4 x^5^ 3 x^4 5 x^3 2 x^2 x 3) ( x^5^ 7 x^4 2 x^3 x^2 3 x 4)
16. Consider P ( x )= 2 x^5 − 3 x^4 + 3 x^2 − 5 and Q ( x )= x^5 + 6 x^4 − 4 x^3 − x + 7 to find:
a) P ( x )+ Q ( x )= b) P ( x )− Q ( x )=
17. Consider P ( x )= 4 x^3 + 6 x^2 − 2 x + 3 , Q ( x )= 2 x^3 − x + 7 and R ( x )= 7 x^2 − 2 x + 1 to obtain:
a) P ( x )+ Q ( x )+ R ( x )= b) P ( x )− Q ( x )− R ( x )= c) P ( x )+ 3 Q ( x )− 2 R ( x )=
18. Evaluate P ( x ) for each of the values of x indicated:
a) P ( x )= x^2 + 1 , para x = 1
c) P ( x )= x^2 + x + 2 , para x = 2
b) P ( x )= x^3 + 1 , para x =− 1
d) P ( x )=− x^2 − x − 2 , para x =− 2
19. Perform the following products with polynomial expressions:
a) 3 · (a – 5) =
b) – 2 · (6 + 12a) =
c) 11 · (2a – 3) =
d) – 7 · (3 + 20b) =
e) 4x · (– 12x + 6) =
f) a^2 · (a – 12) =
g) – 3a · (a^2 – 6a – 7) =
h) 2x^2 · (–13x – 14) =
i) (15x + 4) · (– 3) =
j) (60x + 40) · (– 300) =
k) (2x^3 – 3x^2 +5x – 1 )· (2x + 3)
l) (^) 2
2 x 1 =
m) (^) 2 2
x 3 x 2 =
20. Multiply the following couples of polynomials:
a) P ( x )= 3 x^2 + 2 x − 3 y Q ( x )= x − 2
b) P ( x )= 2 x
2
− 3 x − 2 y Q ( x )= x
2 − 1
c) P ( x )= x
(^5) − 2 x (^4) + 1 2 x^3 − 3 5
y Q ( x^ )= x^
(^4) −^7 4 x^3 + 2 3 x^2 − 3 7 x
d) P ( x )= x
(^4) + 3 x (^2) + 3 5 x − 4 3
y Q ( x^ )=^2 x
(^3) − x (^2) + 4 3
21. Consider A ( x )= x^2 + 2 x − 2 , B ( x )= x^2 − 3 x + 1 and C ( x )= 2 x − x^2 + 3 , to calculate:
a) A ( x ) · B ( x )=
b) 2 · A ( x ) · B ( x )=
c) B ( x ) · [− C ( x )]=
d) [− A ( x )] · B ( x ) ·C ( x )=
e) [− C ( x )] · A ( x )=
f) A ( x ) · [− B ( x )] ·C ( x )=
22. Perform the following divisions:
a)
6 x ⁴− 8 x ³+ 12 x − 4 2 x ²
c)
24 x ⁵ y ⁴+ 18 x ⁴ y ⁵− 48 x ⁹ y ³ 6 x ² y ³
b)
30 x ⁴− 6 x − 6 x
d)
− 3 x ³ y ⁵+ 6 x ⁵ y ⁷− 9 x ² y ⁶ 3 x ² y ²
23. Use the general method to perform the following divisions of polynomials:
a) (x^2 – 9x – 10) : (x + 1) =
b) (6x^4 + 16x^3 + 11x^2 + 6x + 4) : (3x^2 + 5x + 1) =
c) (4x^5 – 24x^4 + 37x^3 – 16x^2 + 16x + 4) : (x^3 – 4x^2 + 2x – 3) =
d) (x^4 – 9x^2 + x +3) : (x + 3) =
24. Divide using Ruffini's Rule. Clearly indicate the quotient and remainder:
a) (x^4 – 9x^2 + x +3) : (x + 3) =
b) (7x^4 – 16x^3 + 24x – 96) : (x + 5) =
c) (2x^3 + x^2 – 3x + 5) : (x + 1) =
d) (6x^4 – 5x^2 +7x +3): (x + 1) =
25. Divide using Ruffini and clearly indicate quotient and remainder:
a) ( x^4 − 2 x^3 + 3 x^2 + 5 x − 2 ):( x − 2 )=
b) ( 3 x^3 − 5 x^2 + 7 x + 3 ):( x + 3 )=
d) ( x^3 − 6 x^2 + 15 ):( x + 5 )=
e) ( x^3 − 7 x^2 + 8 x − 3 ):( x − 2 )=
