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Practice
Simplifying Polynomial Expressions
Easy
Normal
Difficult
Simplifying Polynomial Expressions: Problems with Solutions
By
Catalin David
Problem 1
Which of these is a monomial,
3x
2
or 5x
-3
?
5x
-3
3x
2
Solution:
3x
2
because the exponent of the variable has to be a nonnegative integer.
Problem 2
Which of these is a monomial,
-4x
3
or 2x
2
- 1?
-4x
3
2x
2
-1
Solution:
-4x
3
because 2x
2
-1
is formed by two unlike monomials.
2x
2
- 1 is a binomial.
Problem 3
Which of these is a monomial,
$4x^{7}$ or $2x^{\frac{1}{3}}$ ?
4x
7
$2x^{\frac{1}{3}}$
Solution:
4x
7
because the exponent of the variable has to be a nonnegative integer.
Problem 4
Which of these is a monomial,
$\dfrac{4}{x^{3}}$ or $ -7x^{5}$?
-7x
5
$\frac{4}{x^3}$
Solution:
$-7x^{5}$ because $ \dfrac{4}{x^{3}}$ =$4x^{-3}$
Exponent of the variable has to be a nonnegative integer.
Problem 5
Are -6x
5
and 2x
5
like terms?
Yes
No
Solution:
Yes, because they have the same variables with the same exponent.
Problem 6
Are 4x
3
and 4x
7
like terms?
Yes
No
Solution:
No, because the exponents of the variables are not the same.
Problem 7
Are $2x^{7}$ and $ 2y^{7}$ like terms?
Yes
No
Solution:
No, because they don't have the same variable.
Problem 8
Are -8x
2
y and 7x
2
y like terms?
Yes
No
Solution:
Yes, because they have the same variables with the same exponent.
Problem 9
Are $\dfrac{2}{3}ab^{3}y$ and $\dfrac{4}{7}a^{3}b$ like terms?
Yes
No
Solution:
No, because the variables don't have the same exponents.
Problem 10
6x + 8x =
Solution:
6x + 8x = (6 + 8)x = 14x
Problem 11
3x
4
+ 9x
4
=
6x
4
27x
4
12x
12x
4
Solution:
3x
4
+ 9x
4
= (3 + 9)x
4
= 12x
4
Problem 12
6x
5
+ 8x
5
=
14x
14x
5
15x
5
13x
5
Solution:
6x
5
+ 8x
5
= 14x
5
Problem 13
-7x
2
+ 10x
2
=
17x
2
13x
2
3x
2
-17x
2
Solution:
-7x
2
+ 10x
2
= (-7 + 10)x
2
= 3x
2
Problem 14
-12x
3
+ 5x
3
=
-7x
3
17x
3
7x
3
8x
3
Solution:
-12x
3
+ 5x
3
= -7x
3
Problem 15
-8x
6
+ 8x
6
=
Solution:
-8x
6
+ 8x
6
= (-8 + 8)x
6
= 0x
6
= 0
Problem 16
9x
6
- 3x
6
=
5x
6
-6x
6
12x
6
6x
6
Solution:
9x
6
- 3x
6
=(9 - 3)x
6
= 6x
6
Problem 17
x
4
- 8x
4
=
9x
4
-9x
4
7x
4
-7x
4
Solution:
x
4
- 8x
4
= (1 - 8)x
4
= -7x
4
Problem 18
7a
4
- 11a
4
=
15a
4
4a
4
-4a
4
3a
4
Solution:
7a
4
- 11a
4
= -4a
4
Problem 19
-4x
9
- 6x
9
=
-2x
9
-10x
9
10x
9
2x
9
Solution:
-4x
9
- 6x
9
= (-4 - 6)x
9
= -10x
9
Problem 20
-8y
2
- 9y
2
=
-17y
2
17y
2
-y
2
y
2
Solution:
-8y
2
- 9y
2
= -17y
2
Problem 21
4x
3
⋅ 6 =
24x
12x
3
$\frac{2}{3}x^3$
24x
3
Solution:
4x
3
⋅ 6 = 4⋅6x
3
= 24x
3
Problem 22
-7x
8
⋅ (-3)=
-21x
8
-7x
8
21x
21x
8
Solution:
-7x
8
⋅(-3) = -7⋅(-3)x
8
= 21x
8
Problem 23
5x
3
⋅(-2)=
10x
3
-12x
3
-(-10)x
3
-10x
3
Solution:
5x
3
⋅(-2) = -2 ⋅ 5x
3
= -10x
3
Problem 24
8x ⋅ 3x=
24x
$\frac{8}{3}x^2$
24
24x
2
Solution:
8x ⋅ 3x = 8 ⋅ 3x
1 + 1
= 24x
2
Problem 25
-9x
2
⋅ (-4x)=
36x
3
-36x
3
36x
2
-36x
2
Solution:
-9x
2
⋅ (-4x) = (-9) ⋅ (-4)x
2+1
= 36x
3
Problem 26
6x
4
⋅ 3x
2
=
9x
6
18x
6
9x
5
18x
8
Solution:
6x
4
⋅ 3x
2
= 6 ⋅ 3x
4 + 2
= 18x
6
Problem 27
-4x
5
⋅ 2x
4
=
-8x
9
-6x
9
8x
9
-8x
20
Solution:
-4x
5
⋅ 2x
4
= -4 ⋅ 2x
5 + 4
= -8x
9
Problem 28
6x
5
⋅ 8y
4
=
48x
9
y
4
48x
9
y
48x
9
48x
5
y
4
Solution:
6x
5
⋅ 8y
4
= 6 ⋅ 8x
5
y
4
= 48x
5
y
4
Problem 29
-4x
2
⋅ 7y
3
=
28x
2
y
3
-28x
2
y
3
-3x
2
y
3
-28x
2
y
2
Solution:
-4x
2
⋅ 7y
3
= -4 ⋅ 7x
2
y
3
= -28x
2
y
3
Problem 30
48x
3
: (-6)=
-42x
3
8x
-3
8x
3
-8x
3
Solution:
48x
3
: (-6) = [48 : (-6)]x
3
= -8x
3
Problem 31
12x
6
÷ 4x
3
=
3x
2
8x
3
3x
3
4x
3
Solution:
12x
6
÷ 4x
3
= 12 : 4x
6 - 3
= 3x
3
Problem 32
32x
8
÷ 8x =
24x
7
4x
7
4x
4x
8
Solution:
32x
8
÷ 8x = 32 ÷ 8x
8 - 1
= 4x
7
Problem 33
-15x
5
÷ 5x
4
=
Solution:
-15x
5
÷ 5x
4
= (-15 ÷ 5)x
5 - 4
= -3x
1
= -3x
Problem 34
-24x
8
÷ 3x
8
=
Solution:
-24x
8
÷ 3x
8
= (-24 ÷ 3)x
8 - 8
= -8x
0
=-8
Problem 35
(-2x)
4
=
8x
4
-16x
4
-8x
4
16x
4
Solution:
(-2x)
4
= (-2)
4
x
1 ⋅ 4
= 16x
4
Problem 36
(2x
3
)
4
=
16x
3
8x
12
16x
7
16x
12
Solution:
(2x
3
)
4
= 2
4
x
3 ⋅ 4
= 16x
12
Problem 37
(-3x
3
)
3
=
27x
9
-27x
9
-9x
9
9x
9
Solution:
(-3x
3
)
3
= (-3)
3
x
3 ⋅ 3
= -27x
9
Problem 38
5x - 3 + 4x + 7 =
Solution:
5x - 3 + 4x + 7 = 5x + 4x = 3 + 7 = 9x + 4
Problem 39
3x
2
+ 9 + 7x - 4 =
3x
2
+ 7x + 9
3x
2
+ 9 + 3x
3x
2
+ 15x - 4
3x
2
+ 7x + 5
Solution:
3x
2
+ 9 + 7x - 4 = 3x
2
+ 7x + 9 - 4 = 3x
2
+ 7x + 5
Problem 40
-5x
3
+ 8x - 5 - 3x
3
- 4x + 1=
-8x
3
+4x-4
-2x
3
+4x-4
-8x
3
+12x-4
-8x
3
+4x-6
Solution:
-5x
3
+ 8x - 5 - 3x
3
- 4x + 1=
-5x
3
-3x
3
+ 8x - 4x - 5 + 1 =
-8x
3
+ 4x - 4
Problem 41
6x
3
- 4x
2
+ 5x - 1 - 3x
2
+ 4 =
6x
3
- 7x
2
6x
3
- 7x
2
+ 5x
6x
3
+ 7x
2
+ 5x + 3
6x
3
- 7x
2
+ 5x + 3
Solution:
6x
3
- 4x
2
+ 5x - 1 - 3x
2
+ 4 =
6x
3
- 4x
2
- 3x
2
+ 5x - 1 + 4 =
6x
3
- 7x
2
+ 5x + 3
Problem 42
(6x
2
- 5x + 3) - (4x
2
+ 7x + 8) =
2x
2
+ 2x + 11
2x
2
+ 2x - 5
2x
2
- 12x + 5
2x
2
- 12x - 5
Solution:
(6x
2
- 5x + 3) - (4x
2
+ 7x + 8) =
6x
2
- 5x + 3 - 4x
2
- 7x - 8 =
2x
2
- 12x - 5
Problem 43
4 ⋅ (3x - 5) =
Solution:
4 ⋅ (3x - 5) = 4 ⋅ 3x + 4 ⋅ (-5) = 12x - 20
Problem 44
-8⋅(2x
2
- 7x + 3)=
-16x
2
- 7x + 3
16x
2
- 7x + 3
-16x
2
+56x-24
-16x
2
-56x+24
Solution:
-8⋅(2x
2
- 7x + 3)=
-8⋅2x
2
+ (-8)⋅(-7)x + (-8)⋅3=
-16x
2
+ 56x - 24
Problem 45
(-5x
2
+ 4x - 1)⋅6=
-30x
2
+ 24x - 6
-30x
2
+ 4x - 1
-5x
2
+ 4x - 6
-30x
2
- 24x - 6
Solution:
(-5x
2
+ 4x - 1)⋅6 =
6⋅(-5)x
2
+ 6⋅4x + 6⋅(-1) =
-30x
2
+ 24x - 6
Problem 46
3x⋅(2x - 5)=
6x - 15x
6x
2
- 15x
6x
2
- 5
6x
2
+ 15x
Solution:
3x⋅(2x - 5)=3x⋅2x + 3x⋅(-5)=6x
2
- 15x
Problem 47
(-8x
2
+ 7x - 3)⋅9x=
-72x
3
+7x
2
-27x
-72x
3
+ 7x - 3
-8x
2
+ 7x - 27x
-72x
3
+ 63x
2
- 27x
Solution:
(-8x
2
+ 7x - 3)⋅9x=
9x⋅(-8)x
2
+ 9x⋅7x + 9x⋅(-3) =
-72x
3
+ 63x
2
- 27x
Problem 48
-5x
3
⋅(-2x
2
+ 3x - 4) =
10x
5
- 15x
4
+ 20x
3
-10x
5
- 15x
4
+ 20x
3
-7x
5
- 8x
4
- 9x
3
10x
5
- 15x
4
- 20x
3
Solution:
-5x
3
⋅(-2x
2
+ 3x - 4) =
-5x
3
⋅(-2)x
2
- 5x
3
⋅3x - 5x
3
⋅(-4) =
10x
5
- 15x
4
+ 20x
3
Problem 49
2x
2
⋅(-4x
3
+ 2x - 5) =
-8x
5
+ 4x
3
- 10x
2
8x
5
+ 4x
3
- 10x
2
-8x
5
+ 2x - 5
8x
5
+ 4x
3
- 5x
2
Solution:
2x
2
⋅(-4x
3
+ 2x - 5) =
2x
2
⋅(-4)x
3
+ 2x
2
⋅2x + 2x
2
⋅(-5) =
-8x
5
+ 4x
3
- 10x
Problem 50
(2x
2
)
3
(3x - 2)=
24x
7
- 16x
6
24x
7
- 16x
2
18x
7
- 16x
6
24x
7
+ 16x
6
Solution:
(2x
2
)
3
(3x - 2) =
2
3
(x
2
)
3
(3x - 2) =
8x
6
(3x - 2) =
8x
6
⋅3x + 8x
6
⋅(-2)=
24x
7
- 16x
6
Problem 51
(4x - 3)⋅(2x + 7)=
8x
2
- 21
6x
2
+ 33x - 21
8x
2
+ 22x + 21
8x
2
+ 22x - 21
Solution:
(4x - 3)⋅(2x + 7) =
4x⋅2x + 4x⋅7 - 3⋅2x - 3⋅7 =
8x
2
+ 28x - 6x - 21 = 8x
2
+ 22x - 21
Problem 52
$(3x+4)\cdot(2-5x) =$
-14x
-15x
2
- 14x + 8
15x
2
+ 14x - 8
-15x
2
+ 8
Solution:
$(3x+4)\cdot(2-5x) =$
$3x\cdot2 + 3x\cdot(-5x) +4\cdot2 +4\cdot(-5x)=$
$6x -15x^{2} + 8 - 20x =$
$-15x^{2}-14x +8$
Problem 53
$(3x^{2}+8)\cdot(2x-9) =$
6x
3
+ 27x
2
+ 16x + 72
6x
3
- 27x
2
+ 16x - 72
6x
3
- 27x
2
+ 8x - 36
6x
2
- 27x + 16x - 72
Solution:
$(3x^{2}+8)\cdot(2x-9) =$
$ 3x^{2}\cdot2x + 3x^{2}\cdot(-9) +$
$8\cdot2x +8\cdot(-9)= $
$6x^{3}-27x^{2}+16x-72$
Problem 54
$5x^{2}-(4x-3)(3x+2)=$
-6x
2
+ 2x + 4
7x
2
+ x + 6
-7x
2
+ x + 6
-7x
2
+ 4x + 7
Solution:
$5x^{2}-(4x-3)(3x+2)=$
$5x^{2}-(4x\cdot3x+4x\cdot2-3\cdot3x-3\cdot2)=$
$5x^{2}-(12x^{2}+8x-9x-6)=$
$5x^{2}-(12x^{2}-x-6)=$
$5x^{2}-12x^{2}+x+6=$
$-7x^{2}+x+6$
Problem 55
$2x^{2}+(5x+4)(2x-3)=$
12x
2
- 7x - 12
12x
2
+ 7x + 12
6x
2
- 4x - 10
8x
2
- 4x + 8
Solution:
$2x^{2}+(5x+4)(2x-3)=$
$2x^{2}+[5x\cdot2x+5x\cdot(-3)+4\cdot2x+4\cdot(-3)]=$
$2x^{2}+(10x^{2}-15x+8x-12)=$
$2x^{2}+(10x^{2}-7x-12)=$
$2x^{2}+10x^{2}-7x-12=$
$12x^{2}-7x-12$
Problem 56
$3(5x-2)(2x+3)=$
30x
2
+ 33x - 18
10x
2
+ 11x - 6
30x
2
- 18
30x
2
+ 66x - 9
Solution:
$3(5x-2)(2x+3)=3(5x\cdot2x+5x\cdot3-2\cdot2x-2\cdot3) =$
$ 3(10x^{2}+15x-4x-6) =$
$ 3(10x^{2}+11x-6)=$
$30x^{2}+33x-18$
Problem 57
$(4x-3)(3x-7)(-2)=$
24x
2
- 74x + 42
-24x
2
+ 74x - 42
-24x
2
+ 37x - 21
x
2
+ 74x - 42
Solution:
$(4x-3)(3x-7)(-2)=[4x\cdot3x+4x\cdot(-7)-3\cdot3x-3\cdot(-7)](-2) =$
$ (12x^{2}-28x-9x+21)(-2) =$
$ (12x^{2}-37x+21)(-2)=$
$-24x^{2}+74x-42$
Problem 58
(x + 3)
2
=
x
2
+ 9
2x + 6
x
2
+ 3x + 9
x
2
+ 6x + 9
Solution:
We use the formula of the square of a binomial:
$(a+b)^{2}=a^{2}+2\cdot a\cdot b + b^{2}$
$(x+3)^{2}=x^{2}+2\cdot x\cdot3 +3^{2}=x^{2}+6x+9$
Problem 59
$(x+5)^{2}=$
x
2
+ 5x + 25
x
2
+ 25
x
2
+ 10x + 25
x
2
+ 25x + 25
Solution:
We use the formula of the square of a binomial:
$(a+b)^{2}=a^{2}+2\cdot a\cdot b + b^{2}$
$(x+5)^{2}=x^{2}+2\cdot x\cdot5 +5^{2}=x^{2}+10x+25$
Problem 60
$(2x+7)^{2}=$
4x
2
+ 36x + 49
4x
2
+ 28x + 49
4x
2
+ 14x + 49
4x
2
+ 49
Solution:
We use the formula of the square of a binomial:
$(a+b)^{2}=a^{2}+2\cdot a\cdot b + b^{2}$
$(2x+7)^{2}=(2x)^{2}+2\cdot 2x\cdot7 +7^{2}=4x^{2}+28x+49$
Problem 61
$(x-2)^{2}=$
x
2
- 4
x
2
-2x+4
x
2
+ 4x - 4
x
2
- 4x + 4
Solution:
We use the formula of the square of a binomial:
$(a-b)^{2}=a^{2}-2\cdot a\cdot b + b^{2}$
$(x-2)^{2}=x^{2}-2\cdot x\cdot2 +2^{2}=x^{2}-4x+4$
Problem 62
$(x-7)^{2}=$
x
2
- 49
x
2
- 7x + 49
x
2
- 14x + 49
x
2
- 28x + 49
Solution:
We use the formula of the square of a binomial:
$(a-b)^{2}=a^{2}-2\cdot a\cdot b + b^{2}$
$(x-7)^{2}=x^{2}-2\cdot x\cdot7 +7^{2}=x^{2}-14x+49$
Problem 63
$(5x-3)^{2}=$
25x
2
- 9
25x
2
+ 9
2525x
2
- 30x - 9
25x
2
- 30x + 9
Solution:
We use the formula of the square of a binomial:
$(a-b)^{2}=a^{2}-2\cdot a\cdot b + b^{2}$
$(5x-3)^{2}=(5x)^{2}-2\cdot 5x\cdot3+3^{2}=25x^{2}-30x+9$
Problem 64
$(x+5)\cdot(x-5)=$
25 - x
2
x
2
+ 25
x
2
-10x + 25
x
2
- 25
Solution:
We use the product of the sum and the difference of the two terms:
$(a+b)\cdot(a-b)=a^{2}-b^{2}$
$(x+5)\cdot(x-5)=x^{2}-5^{2}=x^{2}-25$
Problem 65
$(x-3)\cdot(x+3)=$
9 - x
2
x
2
- 9
x
2
- 3
x
2
- 3x + 9
Solution:
We use the product of the sum and the difference of the two terms:
$(a+b)\cdot(a-b)=a^{2}-b^{2}$
$(x-3)\cdot(x+3)=x^{2}-3^{2}=x^{2}-9$
Problem 66
$(3x-4)\cdot(3x+4)=$
3x
2
- 4
9x
2
- 16
-9x
2
+ 16
9x - 8
Solution:
We use the product of the sum and the difference of the two terms:
$(a+b)\cdot(a-b)=a^{2}-b^{2}$
$(3x-4)\cdot(3x+4)=(3x)^{2}-4^{2}=9x^{2}-16$
Problem 67
$(5x+7)\cdot(5x-7)=$
25x
2
- 49
-25x
2
+ 49
5x
2
- 49
25x
2
- 28
Solution:
We use the product of the sum and the difference of the two terms:
$(a+b)\cdot(a-b)=a^{2}-b^{2}=$
$(5x+7)\cdot(5x-7)=(5x)^{2}-7^{2}=25x^{2}-49$
Problem 68
$(9-x)\cdot(9+x)=$
x
2
+ 81
81x
2
x
2
- 81
81-x
2
Solution:
We use the product of the sum and the difference of the two terms:
$(a+b)\cdot(a-b)=a^{2}-b^{2}=$
$(9-x)\cdot(9+x)=9^{2}-x^{2}=81-x^{2}$
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