ADDITION OF POLYNOMIALS

Consider the two polynomials given below

(5x² + 4x + 1) and  (2x² + 5x + 2)

We may use any one of the methods to add two polynomials.

(i)  Vertically

(ii)  Horizontally

In vertical form, align the like terms and add as shown below. 

In horizontal form, use the Associative and Commutative properties to regroup like terms together and combine them as shown below.  

Question 1 :

Add

(5x⁴-3x²+4) + (6x³-4x²-7)

Solution :

=  (5x⁴-3x²+4) + (6x³-4x²-7)

Combining the like terms :

=  5x⁴+6x³-3x²-4x²+4-7

=  5x⁴+6x³+(-3x²-4x²)+(4-7)

=  5x⁴+6x³-7x²-3

Question 2 :

Add

(5x³-7x²+3x-4) + (8x³+2x²+3x-7)

Solution :

=  (5x³-7x²+3x-4) + (8x³+2x²+3x-7)

Combining the like terms :

= (5x³+8x³) + (-7x²+2x²)+(3x+3x)+(-4-7)

=  13x³-5x²+6x-11

Question 3 :

Find the perimeter of triangle given below.

Solution :

Perimeter of triangle ABC  =  Sum of length of sides AB, BC, CA

Perimeter of triangle ABC  =  AB + BC + CA

AB  =  12b-8, BC  =  12b-8 and CA  =  9b+8

Perimeter of triangle ABC  =  12b-8+12b-8 +9b+8

=  12b+12b+9b-8-8+8

=  33b - 8

Question 4 :

Find the perimeter of triangle given below.

Solution :

By finding sum of length of all side, we get perimeter of the trapezium.

=  x²-2x+1+x²-2+x+3+3x-4

=  (x²+x²)+(-2x+x+3x)+(1-2+3-4)

=  2x²+2x-2

Question 5 :

The area of a parallelogram is 10x²+16x-8 square units. A circle has a radius of x units. A right triangle has legs of x units and x-1 units. Express the total area of all three shapes in terms of x and π).

Solution :

Area of parallelogram  =  10x²+16x-8  ------(1)

Area of circle  =  πr²

Here radius  =  x units

Area of circle  =  πx²  ------(2)

Area of triangle  =  (1/2) ⋅ base ⋅ height

=  (1/2) ⋅ x(x-1)

=  x²/2 - x/2  ------(3)

(1) + (2) + (3)

=  10x² + 16x - 8 + πx² + x²/2 - x/2

=  (10x²+πx²+x²/2) + 16x-x/2 - 8

=  x²(10+π+1/2) + x(16-1/2) - 8

=  x²[(20+2π+1)/2] + x[(32-1)/2] - 8

=  x²[(21+2π)/2] + x(31/2) - 8

So, the total area of the given shapes is 

x²[(21+2π)/2] + x(31/2) - 8

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