5. Factorization, Algebraic Expression, Manipulations and Applications

Students Learning Outcomes:

After studying this unit, the students will be able to:

  •  Know that a rational expression behaves like a rational number.
  •  Define a rational expression as the quotient ( ) ( ) p x q x of two polynomials p(x) and q(x) where q(x) is not the zero polynomial.
  • Examine whether a given algebraic expression is a • polynomial or not, • rational expression or not.
  •  Define ( ) ( ) p x q x as a rational expression in its lowest terms if p(x) and q(x) are polynomials with integral coefficients and having no common factor.
  •  Examine whether a given rational algebraic expression is in lowest from or not.
  • Reduce a given rational expression to its lowest terms.
  •  Find the sum, difference and product of rational expressions.
  • Divide a rational expression with another and express the result in it lowest terms.
  •  Find value of algebraic expression for some particular real number.
  • Know the formulas (a + b) 2 + (a – b) 2 = 2(a2 + b2), (a + b) 2 – (a – b) 2 = 4ab
  • Find the value of a2 + b2 and of ab when the values of a + b and a – b are known.
  •  Know the formulas (a + b + c) 2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
  •  find the value of a2 + b2 + c2 when the values of a + b + c and ab + bc + ca are given.
  •  find the value of a + b + c when the values of a2 + b2 + c2 and ab + bc + ca are given.
  • find the value of ab + bc + ca when the values of a2 + b2 + c2 and a + b + c are given.
  •  know the formulas (a + b) 3 = a3 + 3ab(a + b) + b3, (a - b) 3 = a3 - 3ab(a - b) - b3,
  •  find the value of a3 ± b3 when the values of a ± b and ab are given
  •  find the value of x3 ± when the value of x ± is given.
  •  know the formulas a3 ± b3 = (a ± b)(a2 ± ab + b2).
  • • find the product of 1 x x + and 2 2 1 x 1. x + - • find the product of 1 x x - and 2 2 1 x 1. x + + • find the continued product of (x + y) (x - y) (x2 + xy + y2 ) (x2 - xy + y2 ).
  •  recognize the surds and their application.
  • explain the surds of second order.
  • Use basic operations on surds of second order to rationalize the denominators and evaluate it.
  • explain rationalization (with precise meaning) of real numbers of the types 1 1 , a bx x y + + and their combinations where x and y are natural numbers and a and b integers.
  • Introduction
  •  Algebraic Expressions Algebra is a generalization of arithmetic. Recall that when operations of addition and subtraction are applied to algebraic terms, we obtain an algebraic expression. For instance, 5x2 - 3x + 2 x and 3xy + 3 x (x ≠ 0) are algebraic expressions. Polynomials A polynomial in the variable x is an algebraic expression of the form P(x) = anxn + an-1xn-1 + an-2xn-2 + …+ a1x + a0, an ≠ 0 …… (i) ,
  • For detailed content consult PTB 9th Science Book Unit 4 "Algebraic Expression and Algebraic Formulas". 

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