3

ALGEBRA
Percent increase or decrease
 Denoting:  base (x), amount (y), percent (p) and where (x + y) denotes an original or base value x increased (+) or decreased (- ) by amount y, using proportions x : 100 = (x + y) : (100 + p)     and    y : p = (x + y) : (100 + p) Example:  In the price of \$33 included is tax of 10%. What is net price? Example:  A price reduced by 20% amounts \$330, how much is reduced? Polynomial and/or polynomial functions and equations
A polynomial (in a variable x) is a function or an expression written in standard form
f (x) = anxn + an -1xn -1 + an -2xn -2 + . . . + a2x2 + a1x + a0
where,  an, an -1, . . . , a2, a1, a0  are real numbers called coefficients. If an is non-zero, polynomial is of degree n (n is a natural number).
If    g (x) = bmxm + bm -1xm -1 + . . . + b2x2 + b1x + b0   then for m = n,
f (x) + g (x) = (an + bn)xn + (an -1 + bn -1)xn -1 + . . . + (a2 + b2)x2 + (a1 + b1)x + (a0 + b0)
Example:  Given  f (x) = - x3 + 2x2 - 3x +    and    g (x) = 2x4 - 5x3 + x - 4  then
f (x) - g (x) = (0 - 2)x4 + (-1 - 5)x3 + (2 - 0)x2 + (- 3 - 1)x + (1 + 4)
= - 2x4 + 4x3 + 2x2 - 4x + 5
Multiplication of polynomials

We multiply polynomials using the distributive law and the commutative and associative laws of multiplication and addition.

 Example:  Given are polynomials,  f (x) = - 2x3 + 4x - 1   and   g (x) = x2 + 3x - 4  then f (x) × g (x) = (- 2x3 + 4x - 1) × (x2 + 3x - 4) = - 2x5 + 4x3 - x2 - 6x4 + 12x2 - 3x + 8x3 - 16x + 4 = - 2x5 - 6x4 + 12x3 + 11x2 - 19x + 4
Division of polynomials
 Dividing two polynomials,  p (x) and  q (x), obtained is quotient Q (x) and remainder R (x) related by p (x) = Q (x) × q (x) + R (x). The two polynomials are divisible if  R (x) = 0.
 Example: Given,  f (x) = 3x3 - 10x2 + 11x - 4 and g (x) = x2 - 2x  + 1, find  Q (x) =  f (x) ¸ g (x). therefore,   3x3 - 10x2 + 11x - 4 = (3x -  4) × (x2 - 2x  + 1)  and   R (x) = 0.
 Example:  Given,  f (x) = x3  + 2x  and  g (x) = - x2  + x  + 1,  find quotient  f (x) ¸ g (x). thus,   x3 + 2x = (- x - 1) × ( - x2 + x  + 1) + 4x  + 1  i.e.,  Q (x) = - x - 1  and  R (x) = 4x  + 1.   Contents A 