WORKSHEET ON ALGEBRAIC IDENTITIES

About "Worksheet on Algebraic Identities"

Worksheet on Algebraic Identities :

Worksheet given in this section is much useful to the students who would like to practice problems on algebraic identities.

Before look at the worksheet, if you would like to know the stuff related to algebraic identities,

Worksheet on Algebraic Identities - Problems

Problem 1 :

Expand :

(5x + 3)

Problem 2 :

If a - b = 3 and a2 + b2 = 29, find the value of ab.

Problem 3 :

Find the value :

[√2 + 1/√ 2]2

Problem 4 :

If (x + a) (x + b) (x + c)  =  x³ - 10 x² + 45 x - 15, then find the value

a² +  b² + c²

Problem 5 :

If 2x + 3y  =  13 and xy  =  6, then find the value of

8x3 + 27y3

Problem 6 :

Factor :

27x3 + 64y3 Worksheet on Algebraic Identities - Solutions

Problem 1 :

Expand :

(5x + 3)

Solution :

(5x + 3)is in the form of (a + b)2.

Instead of a we have 5x and instead of b we have 3.

So, we have to apply the following algebraic identity  to expand (5x + 3)2

(a + b)=  a2 + 2ab + b2

Substitute a  =  5x and b  =  3.

(5x + 3)2  =  (5x)2 + 2 ⋅ 5x ⋅ 3 + 32

(5x + 3)2  =  25x2 + 30x + 9

Problem 2 :

If a - b = 3 and a2 + b2 = 29, find the value of ab.

Solution :

To find the value of ab, we can use the following algebraic identity.

(a - b)2  =  a2 - 2ab + b2

or

(a - b)2  =  a2 + b2 - 2ab

Substitute a - b  =  3 and a2 + b2  =  29.

32  =  29 - 2ab

9  =  29 - 2ab

Add 2ab to each side.

9 + 2ab  =  29

Subtract 9 from each side.

2ab  =  20

Divide each side by 2.

ab  =  20

Problem 3 :

Find the value :

[√2 + 1/√ 2]2

Solution :

[√2 + 1/√ 2]2 is in the form of (a + b)2.

Instead of a we have √2 and instead of b we have 1/√2.

So, we have to apply the following algebraic identity  to find the value of [√2 + 1/√ 2]2

(a + b)2  =  a2 + 2ab + b2

Substitute a  =  √2 and b  =  1/√2.

[√2 + 1/√ 2]2  =  (√2)2 + 2 ⋅ √2 ⋅ 1/√2 + (1/√2)2

Simplify.

[√2 + 1/√ 2]2  2 + 2 ⋅ 1 +  1 / 2

[√2 + 1/√ 2]2  =  2 + 2 +  1 / 2

[√2 + 1/√ 2]2  =  4 + 1 / 2

[√2 + 1/√ 2]2  =  9 / 2

Problem 4 :

If (x + a) (x + b) (x + c)  =  x³ - 10 x² + 45 x - 15, then find the value

a² +  b² + c²

Solution :

To find the value of ab, we can use the following algebraic identity.

(a + b + c)2  =  ab+ c+ 2(ab + bc + ac)

or

a+ b+ c+ 2(ab + bc + ac)  =  (a + b + c)2

Subtract 2(ab + bc + ac) from each side.

a+ b+ c =  (a + b + c)2 - 2(ab + bc + ac) ----(1)

To find the value of a² +  b² + c², we have to know the value of (a + b + c) and (ab + bc + ac).

Let us find the values of (a + b + c) and (ab + bc + ac).

(x + a)(x + b)(x + c  =  x3 + (a+b+c)x2 + (ab+bc+ca)x + abc

x+ (a+b+c)x+ (ab+bc+ca)x + abc  =  x3 - 10x2 + 45x - 15

Compare the coefficients of x2 and x.

Coefficient of x2 :

a + b + c  =  - 10

Coefficient of x :

ab + bc + ca  =  45

(1)----> a+ b+ c =  (a + b + c)- 2(ab + bc + ac)

Substitute.

a+ b+ c =  (-10)- 2(45)

a+ b+ c =  100 - 90

a+ b+ c =  10

Problem 5 :

If 2x + 3y  =  13 and xy  =  6, then find the value of

8x3 + 27y3

Solution :

We can write 8x3 + 27y3 as follows.

8x3 + 27y =  (2x)3 + (3y)3

To find the value of 8x3 + 27y3, we can use the following algebraic identity.

a3 + b3  =  (a + b)3 - 3ab(a + b)

Substitute a  =  2x and b  =  3y.

(2x)3 + (3y)3  =  (2x + 3y)3 - 3 ⋅ 2x ⋅ 3y(2x + 3y)

8x3 + 27y3  =  (2x + 3y)3 - 18xy(2x + 3y)

Substitute 2x + 3y  =  13 and xy  =  6.

8x3 + 27y3  =  (13)3 - 18 ⋅ 6 ⋅ (13)

8x3 + 27y3  =  2197 - 1404

8x3 + 27y3  =  793

Problem 6 :

Factor :

27x3 + 64y3

Solution :

We can write 27x3 + 64yas follows.

27x3 + 64y =  (3x)3 + (4y)3

To factor 27x3 + 64y3, we can use the following algebraic identity.

a3 + b3  =  (a + b)(a2 - ab + b2)

Substitute a  =  3x and b  =  4y.

(3x)3 + (4y)3  =  (3x + 4y)[(3x)2 + 3x ⋅ 4y + (4y)2]

27x3 + 64y3  =  (3x + 4y)(9x2 + 12xy + 16y2) After having gone through the stuff given above, we hope that the students would have understood, "Worksheet on Algebraic Identities".

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 