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Evaluation And Equivalent Expressions Of Algebraic Expressions

Types of Algebraic Expressions

  • Algebraic expressions are classified on the basis of number of terms.

 Various types of algebraic expressions are as follows:

  • Monomial: An expression which has only one term is called monomial.

Eg. \(4x,\; 3y,\; 6z\), etc.

  • Binomial: An expression which has two terms is called binomial.

Eg. \(x^2+4x,\;y+4,\;3x+4\), etc.

  • Trinomial: An expression which has three terms is called trinomial.

Eg. \(x^3+2x+1,\;a^2+b^2+2ab\), etc.

  • Quadrinomial: An expression which has four terms is called quadrinomial.

Eg. \(x^2-y^2+6+x,\;x^3+3+6x^2+2x\), etc.

  • Polynomial: An expression which has two or more than two terms is called polynomial.

Eg. \(6x^3+x^5+x^4+x^2+x+1\), etc.

Illustration Questions

Which one of the following is NOT a polynomial?

A \(2x+3\)

B \(3x\)

C \(3x+5+y\)

D \(1+2x+3y+4z\)

×

A polynomial has two or more than two terms.

Thus, \(3x\) is not a polynomial.

Hence, option (B) is correct.

Which one of the following is NOT a polynomial?

A

\(2x+3\)

.

B

\(3x\)

C

\(3x+5+y\)

D

\(1+2x+3y+4z\)

Option B is Correct

Finding Equivalent Expressions

Equivalent Expressions

  • If two expressions are equal, they are called equivalent expressions.
  • To find the equivalent expression, keep the following rules in mind:
  1. Commutative Property of Addition: On changing the order of numbers, their sum does not change. This is known as commutative property of addition, i.e.

 \(a+b=b+a\)

      2. Commutative Property of Multiplication: On changing the order of numbers, their product does not change. This is known as commutative property of multiplication, i.e.

 \(a×b=b×a\) 

      3. Associative Property of Addition: On changing the grouping of numbers, their sum does not change. This is known as associative property of addition, i.e.

\(a+(b+c)=(a+b)+c\)  

      4. Associative Property of Multiplication: On changing the grouping of numbers, their product does not change. This is known as associative property of multiplication, i.e.

\(a×(b×c)=(a×b)×c\)

      5. Distributive Property of Multiplication over Addition: The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the product together, i.e.

\(a(b+c)=a×b+a×c\)

      6. Distributive Property of Multiplication over Subtraction: The distributive property of multiplication over subtraction is like the distributive property over addition. Either we find out the difference first and then multiply, or we first multiply with each number and then subtract, the result will always be same, i.e.

\(a(b-c)=ab-ac\)

      7. Like terms can be added and subtracted.

Illustration Questions

Which is the equivalent term of \(5(6+x)\)?

A \(5(6x)\)

B \(30+5x\)

C \(6(5+x)\)

D \(30x\)

×

Given: \(5(6+x)\)

 

Here, distributive property of multiplication over addition can be used.

\([a(b+c)=ab+ac]\)

Thus, the equivalent term of \(5(6+x)\) is

\(5(6+x)=5×6+5x\)

\(5(6+x)=30+5x\)

Hence, option (B) is correct.

Which is the equivalent term of \(5(6+x)\)?

A

\(5(6x)\)

.

B

\(30+5x\)

C

\(6(5+x)\)

D

\(30x\)

Option B is Correct

Evaluating an Expression for Single Variable

  • 'Evaluating an expression' means simplifying an expression down to a single numerical value.
  • An expression can be evaluated by putting the given value of the variable in the expression and then solving it.
  • Here, we will learn how to evaluate an expression when the value of the variable is given.

For example: 

Evaluate: \(4s^2\) when \(s=2\)

Here, the variable is \(s\) and its value is given.

Thus, to evaluate, we simply put \(2\) in place of \(s\).

\(4s^2\)

\(=4(2)^2\)

\(=4×4\)

\(=16\)

Illustration Questions

Evaluate ​​\(a^3\) when \(a =3\).

A \(13\)

B \(42\)

C \(27\)

D \(12\)

×

Given expression: \(a^3\)

Here, the variable is \(a\) and its value is given.

Thus, to evaluate, we simply put \(3\) in place of \(a\).

\(a^3\)

\(=3^3\)

\(=3×3×3\)

\(=27\)

Hence, option (C) is correct.

Evaluate ​​\(a^3\) when \(a =3\).

A

\(13\)

.

B

\(42\)

C

\(27\)

D

\(12\)

Option C is Correct

Illustration Questions

 What will be the equivalent expression of  \(6x(x+y+xy)\)?

A \(6x^2+6xy+6x^2y\)

B \(xy+xy+6x\)

C \(6x+6y+6xy\)

D \(x+y+xy\)

×

Given expression: \(6x(x+y+xy)\)

We will use distributive property to find the equivalent expression.

\(6x(x+y+xy)\)

\(=6x^2+6xy+6x^2y\)

Hence, option (A) is correct.

 What will be the equivalent expression of  \(6x(x+y+xy)\)?

A

\(6x^2+6xy+6x^2y\)

.

B

\(xy+xy+6x\)

C

\(6x+6y+6xy\)

D

\(x+y+xy\)

Option A is Correct

Evaluating an Expression for Two Variables

  • 'Evaluating an expression' means simplifying an expression down to a single numerical value.
  • An expression can be evaluated by putting the given value of variables in the expression and then solving it.
  • Here, we will learn how to evaluate an expression when values of two variables are given.

For example: 

Evaluate: \(a^2b\)  for  \(a = - 2,\,b=3\)

Here, the variables are \(a\) and \(b\) and their values are given.

Thus to evaluate, we simply put \(-2\) and \(3\) in place of \(a\) and \(b\), respectively.

\(a^2b\)

\(=(-2)^2(3)\)

\(= 4×3\)

\(=12\)

Illustration Questions

Evaluate: \((b+d)^2\) for \(b=3, \,d=4\)

A \(49\)

B \(36\)

C \(46\)

D \(50\)

×

Given expression: \((b+d)^2\)

Here, the variables are \(b\) and \(d\) and their values are given.

Thus to evaluate, we simply put \(3\) and \(4\) in place of \(b\) and \(d\), respectively.

\((b+d)^2\)

\(=(3+4)^2\)

\(=(7)^2\)

\(=49\)

Hence, option (A) is correct.

Evaluate: \((b+d)^2\) for \(b=3, \,d=4\)

A

\(49\)

.

B

\(36\)

C

\(46\)

D

\(50\)

Option A is Correct

Evaluating a Polynomial Expression

  • Here, we will learn how to evaluate a polynomial expression when the value of the variable (or variables) is given.

For example: 

Evaluate: \(x^2+2xy+y^2\)  for  \(x=1,\,y=2\)

Here, two variables are used and their values are given.

Thus to evaluate, we simply put the values in place of variables.

\(x^2+2xy+y^2\)

\(=(1)^2+2(1)(2)+(2)^2\)

\(=1+2×2+4\)

\(=1+4+4\)

\(=9\)

Illustration Questions

Evaluate: \(x^4+3x^3-x^2+6\) for \(x=-3\)  

A \(-2\)

B \(-3\)

C \(-1\)

D \(3\)

×

Given expression: \(x^4+3x^3-x^2+6\)

 

To evaluate, we simply put the value in place of the variable. 

    \(x^4+3x^3-x^2+6 \)                    \([ x=-3]\)

\(= (-3)^4+3(-3)^3-(-3)^2+6\)

\(=(-3)×(-3)×(-3)×(-3)+3×(-3)(-3)(-3)-(-3)(-3)+6\)

\(=81-81-9+6\)

\(=-9+6\)

\(=-3\)

Hence, option (B) is correct.

Evaluate: \(x^4+3x^3-x^2+6\) for \(x=-3\)  

A

\(-2\)

.

B

\(-3\)

C

\(-1\)

D

\(3\)

Option B is Correct

Writing Expressions Involving Multiple Operations

  • We know that an expression can have multiple operations.
  • Here, we will learn to write expressions involving multiple operations.

For example: 

Jillian's father is thrice as old as Jillian. Two times the sum of their ages is the age of Jillian's grandfather. Write the expression for the grandfather's age.

Here, we first assume that Jillian's age is \(x\) years. Then, her father's age will be \(3x\) years.

Two times the sum of their ages \((x+3x)\) is grandfather's age.

Thus, grandfather's age is \(2(x+3x)\).

Illustration Questions

An electrician charges \($45\) per hour and spends \($20\) a day on gasoline and \($10\) on grocery. Write an algebraic expression to represent his savings for one day?

A \(45a-20-10\)

B \(10a-45-20\)

C \(20a-45-10\)

D \(45a\)

×

Let 'a' represents the number of hours the electrician works in one day.

He charges \($45\) per hour.

Thus, total money earned by him = \(45a\)

He spends \($20\) a day on gasoline and \($10\) on grocery. That means we need to subtract these amounts from his earnings. 

Thus, his savings for one day is \(45a-20-10\).

Hence, option (A) is correct.

An electrician charges \($45\) per hour and spends \($20\) a day on gasoline and \($10\) on grocery. Write an algebraic expression to represent his savings for one day?

A

\(45a-20-10\)

.

B

\(10a-45-20\)

C

\(20a-45-10\)

D

\(45a\)

Option A is Correct

Illustration Questions

What will be the equivalent expression of  \(a×a+2a×b+3(b+c)\)?

A \(3(b+c)\)

B \(a^2 + 2ab\)

C \(a^2+2ab+3b+3c\)

D \(a^2+2ab+3b\)

×

Given expression: \(a×a+2a×b+3(b+c)\)

We will simplify it to find the equivalent expression by using distributive property. 

\(a×a + 2a×b+3(b+c)\)

\( = a^2 + 2ab+3b+3c\)

Hence, option (C) is correct.

What will be the equivalent expression of  \(a×a+2a×b+3(b+c)\)?

A

\(3(b+c)\)

.

B

\(a^2 + 2ab\)

C

\(a^2+2ab+3b+3c\)

D

\(a^2+2ab+3b\)

Option C is Correct

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