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# Ex.2.5 Q16 Polynomials Solution - NCERT Maths Class 9

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## Question

What are the possible expressions for the dimensions of the cuboids whose volume are given below?

(i). Volume:  $$3 x^{2}-12 x$$

(ii). Volume: $$12 k y^{2}+8 k y-20 k$$

## Text Solution

Reasoning:

(i) Volume of a cubiod = length $$\times$$ breadth $$\times$$ height

What is known?

Volume of cubiod.

What is unknown?

Length, breadth and height of the cuboid.

Steps:

Volume of a cubiod = length $$\times$$ breadth $$\times$$ height

Hence we shall express the given polynomial as product of three expression.

$$3 x^{2}-12 x=3 x(x-4)$$

Length $$= 3$$, breadth $$= x$$, height $$= x-4$$

Length $$= 3$$, breadth $$= x-4$$, height $$= x$$

Length $$= x$$, breadth $$= 3$$, height $$= x-4$$

Length $$= x-4$$, breadth $$= x-4$$, height $$= 3$$

Length $$= x-4$$, breadth $$= 3$$, height $$= x$$

Length $$= x-4$$, breadth $$= 3$$, height$$= x$$

(ii)

What is known?

Volume of cubiod.

What is unknown?

Length, breadth and height of the cuboid.

Steps:

Volume of a cubiod = length $$\times$$ breadth $$\times$$ height

Hence, we shall express the given polynomial as product of three factors

$$12 k y^{2}+8 k y-20 k=4 k\left(3 y^{2}+2 y-5\right)$$

Now taking $$3 y^{2}+2 y-5$$,

find$$2$$  numbers $$p, q$$ such that:

i.  $$p+q=$$co-efficient of  $$y$$

ii .$$p q=$$co-efficient of  $$y^{2}$$ and the constant term.

$$p+q=2$$(co-efficient of y)
$$p q=3 \times-5=-15$$(co-efficient of  $$y^{2}$$ and the constant term.)

By trial and error method, we get

$$p = 5, q = -3.$$

Now splitting the middle term of the given polynomial,

\begin{align} 3 y^{2}+2 y-5 &=3 y^{2}+5 y-3 y-5 \\ &=3 y^{2}-3 y+5 y-5 \\ &=3 y(y-1)+5(y-1) \\ &=(3 y+5)(y-1)\end{align}
Volume $$=4 k(y-1)(3 y+5)$$

Length $$=4 k$$, breadth $$=y-1$$, height $$=3 y+5$$

Length $$=4 k$$, breadth $$=3 y+5$$, height$$=y-1$$

Length $$=y-1$$, breadth $$=4 k$$, height $$=3 y+5$$

Length $$=y-1$$, breadth $$=3y+5 ,$$ height $$=4 k$$

Length $$=3 y+5$$, breadth $$=4 k$$, height $$=y-1$$

Length $$=3 y+5$$, breadth$$=y-1$$ , height $$=4 k$$

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