To learn this concept, first we should be aware of operation in polynomials.
Adding and subtracting polynomials is nothing but combining the like terms.
When we multiply two polynomials, we will follow the order given below.
(1) Signs (2) Number (3) Variable
Let us see how it works,
Multiply (5x2) and (-2x3)
= (5x2) x (-2x3)
= -10x2+3
= -10x5
Find the area A of the shaded regions if all sides are given in m:
Example 1 :
Solution :
Area of shaded region = Area of rectangle
length = 7x and width = 4x
Area of rectangle = 7x(4x)
= 28x2
So, area of shaded region is 28x2 square meter.
Example 2 :
Solution :
Area of triangle = (1/2) ⋅ base ⋅ height
Base = b+8 and height = 3b
= (1/2) ⋅ (b+8) ⋅ 3b
= (1/2) (3b2+24b)
= 3b2/2+24b/2
= 3b2/2+12b
Example 3 :
Solution :
Area of shaded region
= Area of large rectangle - Area of small rectangle
Large rectangle :
length = 14, width = x + 6
Area of large rectangle = 14(x+6)
= 14x + 84 ---(1)
Small rectangle :
length = 5, width = x
Area of large rectangle = 5(x) ==> 5x ---(2)
(1) - (2)
Area of shaded region = 14x + 84 - 5x
= 9x+84
So, area of shaded region is (9x+84) square meter.
Example 4 :
Solution :
Area of trapezium = (1/2) ⋅ h ⋅ (a+b)
h = height and a, b are parallel sides.
h = a+6, a = 6a and b = 3a
Area of trapezium = (1/2) ⋅ (a+6) ⋅ (6a+3a)
= (1/2) ⋅ (a+6) ⋅ (9a)
= (1/2) ⋅ (9a2+54a)
= (9a2/2) + (54a/2)
= (9a2/2) + 27a
So, the area of shaded region is (9a2/2) + 27a square meter.
Example 5 :
Solution :
Area of shaded region
= Area of large rectangle - Area of small rectangle
Larger rectangle :
length = x + 15 and width = 3x + 2
Area of large rectangle = (x+15) (3x+2)
= 3x2+2x+45x+30
= 3x2+47x+30 ------(1)
Small rectangle :
Length = 2x and width = x + 1
Area of large rectangle = (2x) (x+1)
= 2x2+2x ------(2)
(1) - (2)
Area of shaded region = 3x2+47x+30 - (2x2+2x)
= 3x2-2x2+47x-2x+30
= x2+45x+30
So, area of shaded region is (x2+45x+30) square meter.
Example 6 :
Solution :
Area of shaded region
= Area of large rectangle + Area of small rectangle
Large rectangle :
Length = 2z+4, width = z-1
Area of large rectangle = (2z+4)(z-1)
= 2z2-2z+4z-4
= 2z2+2z-4 ------(1)
Small rectangle :
Length = z+2, width = z-1
Area of small rectangle = (z+2)(z-1)
= z2-z+2z-2
= z2+z-2 ------(2)
(1) + (2)
Area of shaded region = 2z2+2z-4 + z2+z-2
Combining like terms, we get
= 3z2+3z-6
So, area of shaded region is 3z2+3z-6 square meter.
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