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) \(\begin{align} \text{Volume of a cubiod} = \text {length} \times \text { breadth } \times \text{height}\end{align}\)

What is known?

Volume of cubiod.

What is unknown?

Length, breadth and height of the cuboid.

Steps:

\[\begin{align} \text{Volume of a cubiod} = \text {length} \times \text { breadth } \times \text{height}\end{align}\]

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:     

\[\begin{align} \text{Volume of cubiod} = \text {length} \times \text { breadth } \times \text{height}\end{align}\]

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.

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

By trial and error method, we get \(p = 5, q = -3.\)

Now splitting the middle term of the given polynomial,

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

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 \(=4 k\)