Question

3.
Find the values of m and n if the following polynomials are perfect square
(i) 36x^4 - 60x^3+ 61x^2- mx +n​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{36x^4-60x^3+61x^2-mx+n}\,\textsf{is a perfect square}$

$\underline{\textsf{To find:}}$

$\textsf{The value of m and n}$

$\underline{\textsf{Solution:}}$

$\textsf{We apply long division square root method}$

$\begin{array}{r|l}&6x^2-5x+3\\\cline{2-2}&\\6x^2&36x^4-60x^3+61x^2-mx+n\\&\\&36x^4\\&\\\cline{2-2}&\\12x^2-5x&****-60x^3+61x^2\\&\\&****-60x^3+25x^2\\&\\\cline{2-2}&\\12x^2-10x+3&**********36x^2-mx+n\\&\\&**********36x^2-30x+n\\&\\\cline{2-2}&****************0****\\\cline{2-2}\end{array}$

$\textsf{Since the given polynomial is a perfect square, we have}$

$\textsf{m=30 and n=3}$

$\underline{\textsf{Answer:}}$

$\textsf{The value of m and n are 30 and 3}$

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Answer 2

SOLUTION :

TO DETERMINE

The value of m and n such that the below polynomial is perfect square

$\sf{}36 {x}^{4} - 60 {x}^{3} + 61 {x}^{2} - mx + n$

EVALUATION

$\sf{}36 {x}^{4} - 60 {x}^{3} + 61 {x}^{2} - mx + n$

$\sf{} = {(6 {x}^{2}) }^{2} - 2.6 {x}^{2} .5x + {(5x) }^{2} + 36 {x}^{2} - mx + n$

$\sf{} = {(6 {x}^{2} - 5x) }^{2} + 36 {x}^{2} - mx + n$

$\sf{} = {(6 {x}^{2} - 5x) }^{2} + 2.3.(6 {x}^{2} - 5x) + 30x- mx + n$

$\sf{} = {(6 {x}^{2} - 5x) }^{2} + 2.3.(6 {x}^{2} - 5x) + {3}^{2} + (30- m)x + (n - 9)$

$\sf{} = {(6 {x}^{2} - 5x + 3) }^{2} + (30- m)x + (n - 9)$

Therefore the last obtained polynomial will be a perfect square if

$\sf{}30 - m \: = 0 \: \: \: and \: \: n - 9 = 0$

Now

$\sf{}30 - m \: = 0 \: \: \:gives \: \: m = 30$

$\sf{}n - 9 = 0 \: \: \: gives \: \: n = 9$

FINAL RESULT

The required value of m and n are 30 and 9 respectively such that the below polynomial is perfect square

$\sf{}36 {x}^{4} - 60 {x}^{3} + 61 {x}^{2} - mx + n$

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