Algebraic Expressions and Identities Test

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Algebraic Expressions and Identities Test - 16

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Weekly Quiz Competition

Question 1
1 / -0

Which of the following expression is equivalent to the expression (y − x + 3) (y + x − 3)?

A
y² − (x − 3)²

B
(y − x)² − 9

C
x² − (y + 3)

D
(x + y)² − 9

Solution

(y − x + 3) (y + x − 3)

= [y − (x − 3)] [y + (x − 3)]

= y² − (x − 3)² [(a − b) (a + b) = a² − b²]

The correct answer is A.
Question 2
1 / -0

What is the value of the expression 9(a² + b) + 2b − 3ab − 3(3a² + 4b)?

A
− b (1 + 3a)

B
a (2 − 3b)

C
6a² − 9ab + 4b

D
18a² − 3ab + 11b

Solution

9(a² + b) + 2b − 3ab − 3(3a² + 4b)

= 9a² + 9b + 2b − 3ab − 9a² − 12b

= (9a² − 9a²) + (9b + 2b − 12b) − 3ab

= 0 − b − 3ab

= − b (1 + 3a)

The correct answer is A.
Question 3
1 / -0

Which of the following pairs of terms are like terms?

A
xyz and lmn

B
3xy² and 9xy²

C
2l²m and 2m²l

D
4x²y²z and 4xy²z²

Solution

The terms having the same algebraic factors are called like terms.

It can be seen that in the terms 3xy² and 9xy², 3xy² has factors 3, x, y, and y; 9xy² has factors 9, x, y, and y.

Thus, the pair of terms 3xy² and 9xy² are like terms.

The correct answer is B.
Question 4
1 / -0

The information in which alternative is correctly matched?

A

Polynomial Coefficient of x² Coefficient of x

7x² + 2x + 3
7 2

B

Polynomial Coefficient of x² Coefficient of x

2x² + 3yz + x
2 0

C

Polynomial Coefficient of x² Coefficient of x

9y² + 6x² + 2yz + 3x
9 3

D

Polynomial Coefficient of x² Coefficient of x

4x² + 2y² + 4z² − 2x
4 2

Solution

The numerical factor of a term is said to be its coefficient.

In the expression 7x² + 2x + 3, the coefficient of x² is 7 and the coefficient of x is 2.

The correct answer is A.
Question 5
1 / -0

Which of the following expressions is equivalent to the expression (z² + 4) (z² − 5)?

A
z⁴ + (5 − 4) z² + 20

B
z⁴ + (4 − 5) z² − 20

C
z² − (5 − 4) z − 20

D
z² − (4 − 5) z + 20

Solution

(z² + 4) (z² − 5)

= (z²)² + [4 + (− 5)] z² + 4 × (− 5) [(x + a) (x + b) = x² + (a + b) x + ab]

= z⁴ + (4 − 5) z² − 20

The correct answer is B.
Question 6
1 / -0

What is the simplified form of the expression (x + y + 6) (x − y + 2)?

A
y² − x² + 2x − 6y + 12

B
x² − y² + 8x − 4y + 12

C
x² + y² + 3xy + x − 2y + 8

D
x² + y²− 2xy + 2x − 3y + 8

Solution

(x + y + 6) (x − y + 2)

= x(x − y + 2) + y(x − y + 2) + 6(x − y + 2)

= x² − xy + 2x + xy − y² + 2y + 6x − 6y + 12

= x² − y² + 8x − 4y + 12

The correct answer is B.
Question 7
1 / -0

What is the product of the expressions (2x + y) and (3x + 2y + z)?

A
4x²+ 3y² + 3xy + 3yz + 3zx

B
6x²+ 4y² + 4xy + yz + 3zx

C
2x²+ 6y² + xy + 7yz + 2zx

D
6x²+ 2y² + 7xy + yz + 2zx

Solution

(2x + y) (3x + 2y + z)

= 2x (3x + 2y + z) + y (3x + 2y + z)

= 6x²+ 4xy + 2xz + 3xy + 2y² + yz

= 6x²+ 2y² + 7xy + yz + 2zx

The correct answer is D.
Question 8
1 / -0

What is the simplified form of the expression (x + y + 2z)(3x − y + z)?

A
3x² − y² + 2z² + 2xy − yz + 7zx

B
3x² + 2y² + 3z² − xy − 2yz + 6zx

C
x² + y² − 2z² + 2xy + 4yz + 6zx

D
x² − y² − 6z² − 3xy + 5yz − 7zx

Solution

(x + y + 2z) (3x − y + z)

= x(3x − y + z) + y(3x − y + z) + 2z(3x − y + z)

= 3x² − xy + zx + 3xy − y²+ yz + 6zx − 2yz + 2z²

= 3x² − y² + 2z² + 2xy − yz + 7zx

The correct answer is A.
Question 9
1 / -0

The expression (l + m) (2m + p) − 2m (p + m) can be simplified as

A
2lm + 2lp − mp − m²

B
lm + 3lp − 2mp − 2m²

C
lm + lp − 2mp

D
2lm + lp − mp

Solution

(l + m) (2m + p) − 2m (p + m)

= l(2m + p) + m(2m + p) − 2m (p + m)

= 2lm + lp + 2m² + mp − 2mp − 2m²

= 2lm + lp − mp

The correct answer is D.
Question 10
1 / -0

The expression (0.4x + 1.2y) (1.5x − 3) + 1.2x − 0.4y can be simplified as

A
0.6x² + 1.2xy + 2.4x

B
0.6x² + 1.8xy − 4y

C
0.6x² + 1.8xy + 2.4x − 4y

D
0.6x² + 1.2xy − 1.2x − 3.2y

Solution

(0.4x + 1.2y) (1.5x − 3) + 1.2x − 0.4y

= 0.4x (1.5x − 3) + 1.2y (1.5x − 3) + 1.2x − 0.4y

= 0.6x² − 1.2x + 1.8xy − 3.6y + 1.2x − 0.4y

= 0.6x² + 1.8xy − 4y

The correct answer is B.
Question 11
1 / -0

What is the product of (a² + 3b) and (a + b)?

A
a³ + 3b² + 3a²b + ab

B
b³ + a² + 3ab² + ab

C
b³ + 3a² + ab² + 3ab

D
a³ + 3b² + a²b + 3ab

Solution

(a² + 3b) (a + b)

= a² × (a + b) + 3b × (a + b)

= (a² × a) + (a² × b) + (3b × a) + (3b × b)

= a³ + 3b² + a²b + 3ab

The correct answer is D.
Question 12
1 / -0

What is the product of (x² − y²) and (3x + 2y)?

A
3x³ + 3y³ + 2x²y − 3xy²

B
2x³ + 2y³ + 3x²y − 2xy²

C
3x³ − 2y³ + 2x²y − 3xy²

D
2x³ − 3y³ − 3x²y + 2xy²

Solution

(x² − y²) (3x + 2y)

= x² × (3x + 2y) − y² × (3x + 2y)

= (x² × 3x) + (x² × 2y) − (y² × 3x) − (y² × 2y)

= 3x³ + 2x²y − 3xy² − 2y³

= 3x³ − 2y³ + 2x²y − 3xy²

The correct answer is C.
Question 13
1 / -0

Which of the following expressions is equivalent to the expression (2x − 3y)² + 24xy?

A
(2x + 3y)²

B
(4x + 9y)²

C
(4x − 9y) (4x + 9y)

D
(2x + 3y) (2x − 3y)

Solution

(2x − 3y)² + 24xy

= (2x)² + (3y)² − 2 × 2x × 3y + 24xy [(a − b)² = a² + b² − 2ab]

= 4x² + 9y² − 12xy + 24xy

= 4x² + 9y² + 12xy

= (2x)² + (3y)² + 2 × 2x × 3y

= (2x + 3y)²

The correct answer is A.
Question 14
1 / -0

What is the value of the expression 3xy (x − 2y + z) − x (2x + xy + yz) + 6xy²?

A
4x²y + 3xy²− xyz − 4x²

B
6x²y + 5xy²+ xyz − 3x²

C
2x²y + 2xyz− 2x²

D
4x²y + 3xyz− 2x²

Solution

3xy (x − 2y + z) − x (2x + xy + yz) + 6xy²

= (3xy × x) + {(3xy) × (−2y)} + (3xy × z) + {(−x) × (2x)} + {(−x) × (xy)}

+ {(−x) × (yz)} + 6xy²

= 3x²y − 6xy² + 3xyz − 2x² − x²y − xyz + 6xy²

= (3x²y − x²y) + (−6xy² + 6xy²) + (3xyz − xyz) − 2x²

= 2x²y + 2xyz− 2x²

The correct answer is C.
Question 15
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The length and breadth of a rectangle are respectively 6pq and 12qr². What is the area of the rectangle?

A
18pq²r²

B
72pq²r²

C
18p²q²r²

D
72p²q²r²

Solution

Length = 6pq

Breadth = 12qr²

The area of the rectangle is given by (length × breadth).

∴ Area of rectangle = 6pq × 12qr²

= (6 × 12) × (pq × qr²)

= 72pq²r²

The correct answer is B.
Question 16
1 / -0

Which of the following expressions is equivalent to the expression 415 × 385?

A
160000 + 225

B
160000 − 225

C
(40)² + 15 × 85

D
(40)² − 15 × 85

Solution

We can write 415 × 385 as (400 + 15)(400 − 15).

The above expression is of the form (a + b) (a − b), where a = 400 and b = 15.

∴ (400 + 15) (400 − 15) = (400)² − (15)²

[(a + b) (a − b) = a² − b²]

= 160000 − 225

The correct answer is B.
Question 17
1 / -0

Which of the following expressions is equivalent to the expression 706 × 695?

A
(600)² + 600 − 6 × 5

B
(700)² + 700 − 30

C
600² + (95 + 6) 600 − 11

D
700² + (95 + 6) 700 − 11

Solution

We can write 706 × 695 as (700 + 6)(700 − 5).

This expression is of the form (x + a) (x + b), where x = 700, a = 6 and b = − 5.

∴ 706 × 695 = (700 + 6) (700 − 5)

= (700)² + (6 − 5) × 700 + 6 × (− 5) [ $∵$ (x + a) (x + b) = x² + (a + b) x + ab]

⇒ 706 × 695 = (700)² + 700 − 30

Hence, the correct answer is option B.
Question 18
1 / -0

From which expression can the expression 14(x − y) − 2(x + xy²) be subtracted to get the expression 5(x + y) − (2x − 7xy²)?

A
2xy² − 4x + y

B
3xy² + 12x − 4y

C
5xy² + 15x − 9y

D
7xy² − 14x + 16y

Solution

The required expression can be obtained by adding 14 (x − y) − 2 (x + xy²) and 5 (x + y) − (2x − 7xy²).

∴{14 (x − y) − 2 (x + xy²)} + {5 (x + y) − (2x − 7xy²)}

= 14x − 14y − 2x − 2xy² + 5x + 5y − 2x + 7xy²

= (14x − 2x + 5x − 2x) + (− 14y + 5y) + (− 2xy² + 7xy²)

= 15x − 9y + 5xy²

Thus, the required expression is 5xy² + 15x − 9y.

The correct answer is C.
Question 19
1 / -0

Which of the following expressions is equivalent to the expression 945 × 855?

A
900² − 45²

B
900² + 45²

C
900² + (45 + 55) 900 + (45 × 55)

D
800² + (45 + 55) 800 + 45 × 55

Solution

We can write 945 × 855 as (900 + 45) (900 − 45).

∴ (900 + 45) (900 − 45) = (900)² − (45)²

[(a + b) (a − b) = a² − b²]

The correct answer is A.
Question 20
1 / -0

The information in which alternative is incorrectly matched?

A

Polynomial Type

x² + y²
Binomial

B

Polynomial Type

x + y Binomial

C

Polynomial Type

x³ Trinomial

D

Polynomial Type

xyz Monomial

Solution

An expression that contains one term is called a monomial, that which contains two terms is called a binomial, and that which contains three terms is called a trinomial.

It can be seen that the expression x³ has only one term. Therefore, it is a monomial.

The correct answer is C.
Question 21
1 / -0

What expression is to be subtracted from the expression 4x² + 2y² + 9xy to obtain the expression 9x² − 10y² + 11xy + 12?

A
13x² − 8y² + 20xy + 12

B
12y² − 5x² − 2xy − 12

C
5x² + 8y² − 2xy − 12

D
8y² − 4x² + 4xy + 12

Solution

The required expression can be obtained by subtracting 9x² − 10y² + 11xy + 12 from the expression 4x² + 2y² + 9xy.

∴ [4x² + 2y² + 9xy] − [9x² − 10y² + 11xy + 12]

= 4x² + 2y² + 9xy − 9x² + 10y² − 11xy − 12

= (4x² − 9x²) + (2y² + 10y²) + (9xy − 11xy) − 12

= − 5x² + 12y² − 2xy − 12

= 12y² − 5x² − 2xy − 12

Thus, the required expression is 12y² − 5x² − 2xy − 12.

Hence, the correct answer is option B.

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Polynomial	Coefficient of x²	Coefficient of x
7x² + 2x + 3	7	2

Polynomial	Type
x² + y²	Binomial

Polynomial	Type
x + y	Binomial

Polynomial	Type
x³	Trinomial

Polynomial	Type
xyz	Monomial

Algebraic Expressions and Identities Test - 16

Result

Algebraic Expressions and Identities Test - 16

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Question 1

y2 − (x − 3)2

(y − x)2 − 9

x2 − (y + 3)

(x + y)2 − 9

Solution

Question 2

− b (1 + 3a)

a (2 − 3b)

6a2 − 9ab + 4b

18a2 − 3ab + 11b

Solution

Question 3

xyz and lmn

3xy2 and 9xy2

2l2m and 2m2l

4x2y2z and 4xy2z2

Solution

Question 4

Polynomial Coefficient of x2 Coefficient of x 7x2 + 2x + 3 7 2

Polynomial Coefficient of x2 Coefficient of x 2x2 + 3yz + x 2 0

Polynomial Coefficient of x2 Coefficient of x 9y2 + 6x2 + 2yz + 3x 9 3

Polynomial Coefficient of x2 Coefficient of x 4x2 + 2y2 + 4z2 − 2x 4 2

Solution

Question 5

z4 + (5 − 4) z2 + 20

z4 + (4 − 5) z2 − 20

z2 − (5 − 4) z − 20

z2 − (4 − 5) z + 20

Solution

Question 6

y2 − x2 + 2x − 6y + 12

x2 − y2 + 8x − 4y + 12

x2 + y2 + 3xy + x − 2y + 8

x2 + y2 − 2xy + 2x − 3y + 8

Solution

Question 7

4x2 + 3y2 + 3xy + 3yz + 3zx

6x2 + 4y2 + 4xy + yz + 3zx

2x2 + 6y2 + xy + 7yz + 2zx

6x2 + 2y2 + 7xy + yz + 2zx

Solution

Question 8

3x2 − y2 + 2z2 + 2xy − yz + 7zx

3x2 + 2y2 + 3z2 − xy − 2yz + 6zx

x2 + y2 − 2z2 + 2xy + 4yz + 6zx

x2 − y2 − 6z2 − 3xy + 5yz − 7zx

Solution

Question 9

2lm + 2lp − mp − m2

lm + 3lp − 2mp − 2m2

lm + lp − 2mp

2lm + lp − mp

Solution

Question 10

0.6x2 + 1.2xy + 2.4x

0.6x2 + 1.8xy − 4y

0.6x2 + 1.8xy + 2.4x − 4y

0.6x2 + 1.2xy − 1.2x − 3.2y

Solution

Question 11

a3 + 3b2 + 3a2b + ab

b3 + a2 + 3ab2 + ab

b3 + 3a2 + ab2 + 3ab

a3 + 3b2 + a2b + 3ab

Solution

Question 12

3x3 + 3y3 + 2x2y − 3xy2

2x3 + 2y3 + 3x2y − 2xy2

3x3 − 2y3 + 2x2y − 3xy2

2x3 − 3y3 − 3x2y + 2xy2

Solution

y² − (x − 3)²

(y − x)² − 9

x² − (y + 3)

(x + y)² − 9

6a² − 9ab + 4b

18a² − 3ab + 11b

3xy² and 9xy²

2l²m and 2m²l

4x²y²z and 4xy²z²

Polynomial Coefficient of x² Coefficient of x

7x² + 2x + 3
7 2

Polynomial Coefficient of x² Coefficient of x

2x² + 3yz + x
2 0

Polynomial Coefficient of x² Coefficient of x

9y² + 6x² + 2yz + 3x
9 3

Polynomial Coefficient of x² Coefficient of x

4x² + 2y² + 4z² − 2x
4 2

z⁴ + (5 − 4) z² + 20

z⁴ + (4 − 5) z² − 20

z² − (5 − 4) z − 20

z² − (4 − 5) z + 20

y² − x² + 2x − 6y + 12

x² − y² + 8x − 4y + 12

x² + y² + 3xy + x − 2y + 8

x² + y²− 2xy + 2x − 3y + 8

4x²+ 3y² + 3xy + 3yz + 3zx

6x²+ 4y² + 4xy + yz + 3zx

2x²+ 6y² + xy + 7yz + 2zx

6x²+ 2y² + 7xy + yz + 2zx

3x² − y² + 2z² + 2xy − yz + 7zx

3x² + 2y² + 3z² − xy − 2yz + 6zx

x² + y² − 2z² + 2xy + 4yz + 6zx

x² − y² − 6z² − 3xy + 5yz − 7zx

2lm + 2lp − mp − m²

lm + 3lp − 2mp − 2m²

0.6x² + 1.2xy + 2.4x

0.6x² + 1.8xy − 4y

0.6x² + 1.8xy + 2.4x − 4y

0.6x² + 1.2xy − 1.2x − 3.2y

a³ + 3b² + 3a²b + ab

b³ + a² + 3ab² + ab

b³ + 3a² + ab² + 3ab

a³ + 3b² + a²b + 3ab

3x³ + 3y³ + 2x²y − 3xy²

2x³ + 2y³ + 3x²y − 2xy²

3x³ − 2y³ + 2x²y − 3xy²

2x³ − 3y³ − 3x²y + 2xy²

(2x + 3y)²

(4x + 9y)²

4x²y + 3xy²− xyz − 4x²

6x²y + 5xy²+ xyz − 3x²

2x²y + 2xyz− 2x²

4x²y + 3xyz− 2x²

18pq²r²

72pq²r²

18p²q²r²

72p²q²r²

(40)² + 15 × 85

(40)² − 15 × 85

(600)² + 600 − 6 × 5

(700)² + 700 − 30

600² + (95 + 6) 600 − 11

700² + (95 + 6) 700 − 11

2xy² − 4x + y

3xy² + 12x − 4y

5xy² + 15x − 9y

7xy² − 14x + 16y

900² − 45²

900² + 45²

900² + (45 + 55) 900 + (45 × 55)

800² + (45 + 55) 800 + 45 × 55

Polynomial Type

x² + y²
Binomial

Polynomial Type

x + y Binomial

Polynomial Type

x³ Trinomial

Polynomial Type

xyz Monomial

13x² − 8y² + 20xy + 12

12y² − 5x² − 2xy − 12

5x² + 8y² − 2xy − 12

8y² − 4x² + 4xy + 12