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Polynomials — Maths STD 9 — Question

Maharashtra BoardEnglish MediumSTD 9MathsPolynomials5 Marks
Question
Write down the information in the form of algebraic expression and simplify.
There is a rectangular farm with length $\left(2 a^2+3 b^2\right)$ metre and breadth $\left(a^2+b^2\right)$ metre.
The farmer used a square shaped plot of the farm to build a house. The side of the plot was ( $a 2- b 2$ ) metre.
What is the area of the remaining part of the farm?
✓

Answer

Length of the rectangular farm $=\left(2 a^2+3 b^2\right) m$
Breadth of the rectangular farm $=\left(a^2+b^2\right) m$
Area of the farm $=$ length $\times$ breadth $=\left(2 a^2+3 b^2\right) \times\left(a^2+b^2\right)$
$=2 a^2\left(a^2+b^2\right)+3 b^2\left(a^2+b^2\right)$
$=2 a^2+2 a^2 b^2+3 a^2 b^2+3 b^4$
$=\left(2 a^4+5 a^2 b^2+3 b^4\right) \text { sq. } m \ldots$
The farmer used a square shaped plot of the farm to build a house.
Side of the square shaped plot $=\left(a^2-b^2\right) m$
$\therefore \text { Area of the plot }=(\text { side })^2$
$=\left(a^2-b^2\right)^2$
$\left.=\left(a^4-2 a^2 b^2+b^4\right) \text { sq m....(ii) }\right) \therefore \text { Area of the remaining farm }=\text { Area of the farm }- \text { Area of the plot }$
$=\left(2 a^4+5 a^2 b^2+3 b^4\right)-\left(a^4-2 a^2 b^2+b^4\right) \ldots[\text { From (i) and (ii)] }$
$=2 a^4+5 a^2 b^2+3 b^4-a^4+2 a^2 b^2-b^4$
$=2 a^4-a^4+5 a^2 b^2+2 a^2 b^2+3 b^4-b^4$
$=a^4+7 a^2 b^2+2 b^4$
$\therefore$ The area of the remaining farm is $\left(a^4+7 a^2 b^2+2 b^4\right)$ sq. m.

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