Algebra- Tutorial--polynomial

Algebra

Tutorial

Polynomials

There are monomials and polynomial. An expression like y = 4x is a monomial. In binomial we have two terms: y= (x+a) is binomial. Note here that x has the exponent 1 and a is equal to: a = a.x^0 Here a appears as a constant,but it is alsoa term of the form: x^n ,that is, x raised to the power of n where n is equal to zero. In Algebra, zero is taken as a positive integer.! Then what is a A Polynomial

polynomial? is a sum of monomilas or terms of the form: x^n [x raised to the power 'n'] where n is a postive integer. n' can be zero, 1 2,3 and so on. Note that n can not be negative: Thus y = 1/x +2 is not polynomial y= X^1/2 or y= square root of x cannot be a term in a polynomial. In general, we write: Y= ao +

a1.x +

a2.x^2+

a3.x^3 +

………… anX^n

The degree of a polyomial is the highest power n in the sum. Thus Y= 4x^3 + 2X+3 is of degree 3. Y= 3x^4 + 2x^2 -3 is of degree 4 The numbers before the x variables are called Coefficients" Thus here 3,2,-3 are the coefficients. Note that the coefficients can be negative or fractions. The coefficient of the highest degree term is called leading coefficient.' Now we

are ready to learn

how to manipulate and use

the

the polynomials.

A practical Example The selling price S of an item decreases with demand,D S = a - bD where a and b are constants. Assume that all the demand for the item is met and sold. Then Revenue R = S.D= D.(a-bD) R= aD - bD^2 Here we have apolynomial of degree 2 A quadratic expression.! ` Sum & Difference of Polynomial We can add two polynomials by adding coefficients of like powers: y= 2x^2 + 3x -4 Y = 3x^2 -2

Add these two polynomials: y+Y = 2x^2 +3 x^2 + 3x -4-2 (2+3)x^2 + 3x - 6 5x^2 + 3x -6 Try this: y= -2X^4 + 2x^3 - 3x^2 -4x +2 Y= 3X^4 +2x^2 -3x +1 You can write the like powers one below the other like a tableau:

y+Y=

y= -2x^4+ 2x^3 3x^2 - 4x+2 Y=3X^4 -2x^2-3x 2 -3x +1 (-2+3) X^4+2X^3 + (-3-2)x^2 + (-4-3)+ (2+1)

y+Y=

x^4 + 2x^3

-5 x^2 -7x +3

Difference is obtained by taking the algebraic difference of coefficients of like terms: y = 3x^3 - 2x^2 + 4x -4 Y= 2x^3 +3x^2 - 2x +2 y - Y = (3-2) x^3 + (-2 -3)x^2 + (4 +2)x -4 -2 y - Y = x^3 -5 x^2 +6 x -6 Evaluating a polynomial Find the the variable; 1 Evaluate the polynimial at x =2:

value' of a polynomial for a particular value of

y = 3.x^2 +4x +2 y= 3x2x2 + 4x2 +2 y=22 2 Evaluate the polynomial at x = -1: y= 2x^3 + x^2 - 3x +1 y= 2(-1) + (-1)^2 -3(-1) +1 3 Find the height of a rocket at time t in seconds when t= 4 seconds:

h= -16 t.t + 120 t + 50 = -256 + 480 + 50 = -256 + 530 = 274 feet

1 Volume of a cylinder is:

V = 2x pi x r x h where pi = 3.14 r is radius and height h John makes two cyclinders: onw with h = 4 x radius and another with h = 3 xradius Find the total volume with radius as the variable. Cylinder1: V1 = 2 x 3.14 x 4 x radius x radius V2 = 2 x 3.14 x 3 x radius x radius V1 + V2 = 6.28 ( 4+3 ) radius^2 V = 6.28 x 7 x r ^2 where r is the radius.

2 The density of Ozone over over atwon a towninincanada Canadawas wasmeasured measured in spring in and two autumn seasons:and expressed as quad D(h) = -0.058 h,h + 2.86 h - 24. 24 [autumn] where h is height in km] D (h) = -0.078 h.h + 3.81 h -32.43 [spring] Find the avarage for a year. [Hint; add the two polynomilas and dvide by two, term by term.

For spring and autumn to gether: D D (h (h)) = -0.0136 0.0136 h.hh.h + 6.67 h - 56.67 Average value: D (h) = 0.0068 h.h + 3.38h - 28.33 3 The height of a July 4th rocket is given by: h = (-g/2) t.t + v t + s where h is the height, v-initial velocity and s initial height. The rocket is fired from the roof of building atheight heightof 40100 feetfeet. initial velocity ity = 20 144feet /second g--accleelration due to gravity [from your physic lessons] is 32 ft/s.s height in feet h = -16 t.t + 144 t + 100 Find the height for various times till it reaches the ground. Setting h =0 16t.t - 144t - 100 = 0 4t.t - 36t -25 =0

Now let us write

Height (feet)

Let us from a small table and approximate the solution first: t y = 4.t^2 - 36t 36t25 - 25 1 -57 2 -81 3 -97 4 -105 105 5 -105 100 6 -97 95 7 -81 90 8 -57 85 80 9 -25 75 10 15 t

Height vs time

70 65 60 55 50 45 40 35 30 25

h 1 57 2 81 3 97 4 105 5 105 6 97 7 81 1 2 3 4 5 6 7 8 8 57 Time (seconds) 9 25 Let us plot the curve of h against t What do you see? -- A symmetrical curve..the rocket's height increases and then decreases. What is the maximum height of the rocket? It is between four and five seconds. Let us take t= 4.5 seconds. h = - 16t.t +144t +100 = 424 feet. Product of Polynomials 1 Take two polynomials product

yxz =

y= 3xx -2 z= 2x 2x(3xx -2) 6xxx - 4x

using distributive property

h

9

2 Find the

product: y.z = y.z = y.z= y.z=

y = x+1 z= 2xx - 3x+3 (x+1)(2xx - 3x+3) x(2xx - 3x +3) + 1 ( 2xx - 3x +3) 2x.x.x -3x.x +3x +2xx -3x +3 2xxx-xx+3

We can use the tableau method for long multiplications: 2xx 3x+ 3 x+ 1 ------------------------------------------------------------Multiply first with x: 2xxx 3x.x + 3x Multiply with 1 2xx3x 3 ----------------------------------------------------------------Add the two: 2xxx xx 0 3 --------------------------------------------------------------Try the following: (x-1) ( 2xx + 3x -2) (2x+1) ( 2xxx - 3xx +2) (2x-1) ( xxx +2xx -3x +1) (xx + 2x -3)(2xx =3x -2) Division of Polynomials This may be tricky…let us proceed slowly. Try to simplify the numerator and denominator first ,if you can. Let us a few examples: 1 Simplify

x.x + 5x+6 --------------x +3

Factoring the Nr:

(x+3)(x+2) --------------x+ 3

2 Simplify Y=

(xx - 6x +9) ( x-2) ------------------------(xx - 4) (x-3)(x-3) (x-2) (x+2) (x-2)

Simplify Nr: Simplify Dr:

3 Simplify:

Y=

(x- 3)(x-3)/(x+2)

Y=

xx + 4x +4 ----------------xx-x-6

Y=

(x+2)(x+2) --------------(x-3)(x+2)

Y=

(x+2)/(x-3)

(x+2)

4 Simplify

5 Simplify:

Y=

(x.x -25) ---------------(xx+7x +10)

Y=

(x-5)/(x+2)

Y=

xxx+5xx+6x --------------------(x+3)

Y=

x(xx+5x+6) --------------(x+3)

Y=

(x+5)(x-5) --------------(x+5)(x+2)

Y=

x(x+3)(x+2) ---------------x+3 x(x+2)

Y= Do it Yoursel

Exercises

1 Simplify: 2 Simplify: 3 Simplify

(x+3)(x.x-3x+2) (x-2)(xx =3x -3) (x+3) (x-2) (x.x -x -6)

4 Simplify

(x.x +7x+12) ----------------(x+3)

5 Simplify

(xx +x-12)(x-3) -------------------(xx-6x+9)

6 Simplify

(xxx + 3xx +2x) --------------------(x+2)

7 The deer population in an island follows the equation: p(t) = -t^4 +21t.t +100 where t is in years. Find when the population may become extinct. [Hint: Solve for p(t) = 0] Plot the graph for t upto 6 years.

cular value of

s time

h

6

7

8

conds)

then decreases.

9