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Book Free DemoThe square root of a polynomial using the factorization method can be used only if the degrees of the polynomial are smaller.
The procedure to find the square root of a polynomial using the factorization method is given by:
Step 1: If the given polynomial is a quadratic equation, factorize them by splitting the middle terms or factorize them using algebraic identities.
Step 2: Take the square root of the factorized terms.
Let us understand how to find the square root of a polynomial by the factorization method in the following examples.
Example:
1. Find the square root of 144(x - a)^2(x -b)^{10}.
Solution:
\sqrt{144(x - a)^2(x -b)^{10}} = 12|(x - a)(x -b)^5|
2. Find the square root of (5x - 2y)^2 + 40xy.
Solution:
\sqrt{(5x - 2y)^2 + 40xy} = \sqrt{25x^2 - 20xy + 4y^2 + 40xy}
= \sqrt{25x^2 + 20xy + 4y^2}
= \sqrt{(5x)^2 + 2(5x)(2y) + (2y)^2}
= \sqrt{(5x + 2y)^2}
= |(5x + 2y)|
3. Find the square root of (6x^2 + 7x - 5)(2x^2 + 9x - 5)(3x^2 + 20x + 25)
Solution:
Let us factorize the polynomials.
6x^2 + 7x - 5 = 6x^2 + 10x - 3x - 5
= 2x(3x + 5) - 1(3x + 5)
= (3x + 5)(2x - 1)
2x^2 + 9x - 5 = 2x^2 + 10x - x - 5
= 2x(x + 5) - 1(x + 5)
= (2x - 1)(x + 5)
3x^2 + 20x + 25 = 3x^2 + 15x + 5x + 25
= 3x(x + 5) + 5(x + 5)
= (x + 5)(3x + 5)
\sqrt{(6x^2 + 7x - 5)(2x^2 + 9x - 5)(3x^2 + 20x + 25)} = \sqrt{(3x + 5)(2x - 1)(2x - 1)(x + 5)(x + 5)(3x + 5)}
= \sqrt{(3x + 5)^2(2x - 1)^2(x + 5)^2}
= (3x + 5)(2x - 1)(x + 5)
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