Algebraic Expressions and Identities

Algebraic expressions are expressions formed from the variables and the constants. A variable can take any value. The value of an expression changes with the value chosen for variables it contains.

  A number line has infinite number of points. A variable can take position on number line.

  Expressions containing one, two or three terms are called monomial, binomial and trinomial respectively.

  Any expression having one or more terms is called polynomial.

  A monomial is obtained on multiplying any monomial with another monomial.

  The numerical factor of a term is called its coefficient.

  An identity is a standard equality which is true for all the values of the variables in the equality.

  Few commonly used identities are

        I.   (a + b)² = a² + b² + 2ab

       II.   (a – b)² = a² + b² – 2ab

      III.   (a + b)(a – b) =a² – b²

      IV.   (x + a)(x + b) = x² + (a+b)x + ab

Note – Mainly the above formulas are used to solve all the problems of this chapter.

1.  Find the value of: x² – 1/5 at x= -1.

2.  What is the value of x² + y² – 10 at x = 0 and y = 0?

3.  Find the product of 9a, 4ab and -2a.

4.  Simplify (a + b + c)(a + b - c).

5.  Using identities evaluate: 8.56 * 11.60.

6.  Using identities evaluate: (99)².

7.  Simplify x(2x – 1) + 5 and find its value at x = -2.

8.  Evaluate the value of (95)² using identities.

1.  Which of the following is the numerical coefficient of x²y²?

        I.   0

       II.   1

      III.   x²

      IV.   y²

2.  Which of the following is the numerical coefficient of -5xy?

        I.   5

       II.   -x

      III.   -5

      IV.   -y

3.  pqr is what type of polynomial?

        I.   Monomial

       II.   Binomial

      III.   Trimonial

      IV.   None of these

4.  The value of x² - 5 at x= -1 is-

        I.   -2

       II.   -1

      III.   -4

      IV.   -5

5.  a²-b² is a product of

        I.   (a+b)(a-b)

       II.   (a+b)(a+b)

      III.   (a-b)(a-b)

      IV.   None of these

6.  Which of the following is the value of (x+ 1/x)²?

        I.   x² + 1/x²

       II.   x² - 1/x²

      III.   x² + 1/x² + 2

      IV.   x²+ 1/x² + 2x

7.  Which of the following is obtained by subtracting x²-y² from y² - x²?

        I.   -2(x²-y²)

       II.   -2(x² + y²)

      III.   2(x²+ y²)

      IV.   2(x²- y²)

8.  What degree does x³ - x²y² - 8y²+ 2 have?

        I.   2

       II.   3

      III.   4

      IV.   7

9.  What is the value of 5x²⁵ - 3x³² + 2x^-12 at x=1?

        I.   0

       II.   2

      III.   4

      IV.   None of these

10.  What is the product of (x+a) and (x+b)?

        I.   x²+ (a-b)x + ab

       II.   x² + (a+b)x - ab

      III.   x² + (a+b)x - ab

      IV.   x² + (a+b)x + ab

ANSWERS

        1.    II

        2.    III

        3.    I

        4.    III

        5.    I

        6.    III

        7.    I

        8.    III

        9.     III

        10.  I

Maximum time- 30 minutes

Maximum marks- 20

1.  Add: a + b + ab; b – c + bc and c + a + ac.      (2)

2.  Verify the identity (x + a)(x + b) = x² + (a + b)x + ab for a = 2, b = 3 and x = 4.      (3)

3.  Find the volume of cuboid whose dimensions are (x² – 2); (2x + 4) and (x - 3).      (4)

4.  Write the terms and coefficients of 3 – xy + yz – xz.      (2)

5.  Simplify: (a + b +c)(a + b – c).      (2)

6.  Simplify the expression x(2x-1) + 5 and its value at x = -2.      (3)

7.  Using suitable identities find (xy + 3p)².      (2)

8.  Subtract 5x² – 6y² + 8y – 5 from 7x² – 5xy + 10y² + 5x – 4y.      (2)