ADD SUBTRACT MULTIPLY AND DIVIDE POLYNOMIALS

Example 1 :

Add (6a + 3) and (4a - 2).

Solution :

= (6a + 3) + (4a - 2)

 = 6a + 3 + 4a - 2

= 10a + 1

Example 2 :

Add (5y² + 8y + 3) and (4y² - 5y - 2).

Solution :

= (5y² + 8y + 3) + (4y² - 5y - 2)

= 5y² + 8y + 3 +  4y² - 5y - 2

= 9y² + 3y + 1

Example 3 :

Subtract (6a - 5) from (-8a + 9).

Solution :

= (-8a + 9) - (6a - 5)

= -8a + 9 - 6a + 5

= -14a + 14

Example 4 :

Subtract (2x² + 3x - 5) from (5x² - 4x - 5).

Solution :

= (5x² - 4x - 5) - (2x² + 3x - 5)

= 5x² - 4x - 5 - 2x² - 3x + 5

= 3x² - 7x

Example 5 :

Multiply (3x - 7) and (7x - 3).

Solution :

= (3x - 7)(7x - 3)

= 21x² - 9x - 49x + 21

= 21x² - 58x + 21

Example 6 :

Multiply (3x² - 5) and (2x + 3).

Solution :

= (3x² - 5)(2x + 3)

= 3x²(2x) + 3x²(3) - 5(2x) - 5(3)

= 6x³ + 9x² - 10x - 15

Example 7 :

Multiply (p² + 5p + 2) and (p² + 5p - 2).

Solution :

= (p² + 5p + 2) and (p² + 5p - 2)

Let x = p² + 5q.

= (x + 2)(x - 2)

Using Algebraic Identity a² - b² = (a + b)(a - b),

= x² - 2²

= x² - 4

Substitute x = p² + 5p.

= (p² + 5p)² - 4

Using Algebraic Identity (a + b)² = a² + 2ab + b².

= (p²)² + 2(p²)(5p) + (5p)² - 4

= p⁴ + 10p³ + 25p² - 4

Example 8 :

Divide (6x² - 54) by (x² + 7x + 12).

Solution :

= (6x² - 54)/(x² + 7x + 12)

= 6(x² - 9)/[(x + 3)(x + 4)]

= 6(x² - 3²)/[(x + 3)(x + 4)]

= [6(x + 3)(x - 3)]/[(x + 3)(x + 4)]

= 6(x - 3)/(x + 4)

Example 9 :

Divide (2 - x) by (x² + 4x - 12).

Solution :

= (2 - x)/(x² + 4x - 12)

= (2 - x)/[(x + 6)(x - 2)]

= -(x - 2)/[(x + 6)(x - 2)]

= -1/(x + 6)

Example 10 :

Divide (x⁴ - 16) by (x² + 5x + 6).

Solution :

= (x⁴ - 16)/(x² + 5x + 6)

=  [(x²)² - 4²]/[(x + 2)(x + 3)]

= [(x² + 4)(x² - 4)]/[(x + 2)(x + 3)]

= [(x² + 4)(x² - 2²)]/[(x + 2)(x + 3)]

= [(x² + 4)(x + 2)(x - 2)]/[(x + 2)(x + 3)]

= [(x² + 4)(x - 2)]/(x + 3)

Find the perimeter of the following shapes.

Example 11 :

Solution :

Length = 2x and width = y

Perimeter of rectangle = 2(length + width)

= 2(2x + y)

= 4x + 2y

Example 12 :

Solution :

Length = x and width = y²

Perimeter of rectangle = 2(length + width)

= 2(x + y²)

= 4x + 2y

Example 13 :

What should be added to x³ + 3x²y + 3xy² + y³ to get x³ + y³ ?

Solution :

Let p(x) be the polynomial.

x³ + 3x²y + 3xy² + y³ + p(x) =  x³ + y³ 

p(x) =  x³ + y³ - (x³ + 3x²y + 3xy² + y³)

Distributing the negative sign,

p(x) =  x³ + y³ - x³ - 3x²y - 3xy² - y³

p(x) =  - 3x²y - 3xy²

So, the required polynomial to be added is - 3x²y - 3xy²

Example 14 :

How much is 21a³ - 17a² less than 89a³ - 64a² + 6a + 16?

Solution :

Let p(x) be the polynomial

= 89a³ - 64a² + 6a + 16 - (21a³ - 17a²)

= 89a³ - 64a² + 6a + 16 - 21a³ + 17a²

= 89a³ - 21a³ - 64a² + 17a² + 6a + 16

= 68a³ - 47a² + 6a + 16

Example 15 :

If P = –(x – 2), Q = –2(y + 1) and R = –x + 2y, find a, when

P + Q + R = ax.

Solution :

P + Q + R = ax

Given that,

P = –(x – 2), Q = –2(y + 1) and R = –x + 2y

-(x - 2) - 2(y + 1) - x + 2y = ax

-x + 2 - 2y - 2 - x + 2y = ax

-2x = ax

So, the value of a is -2.

Example 16 :

Arjun bought a rectangular plot with length x and breadth y and then sold a triangular part of it whose base is y and height is z. Find the area of the remaining part of the plot.

Solution :

Area of rectangular plot = length x width

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

Area of remaining part = xy - (1/2) yz

= y(x - (1/2)z)

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