Polynomial are mathematical expressions involving variables raised with non negative integers and coefficients. The complete factorization of the polynomial 49p⁸ + 42p⁴ + 9 is (7p⁴+3)².
Polynomial are mathematical expressions involving variables raised with non negative integers and coefficients(constants who are in multiplication with those variables) and constants with only operations of addition, subtraction, multiplication and non-negative exponentiation of variables involved.
Factorization is expressing a mathematical quantity in terms of multiples of smaller units of similar quantities.
The given polynomial can be factorized as shown below,
49p⁸ + 42p⁴ + 9
Break 42p⁴ into two terms such that the sum of the two terms is 42p⁴, while there product is equal to the product of 49p⁸ and 9,
= 49p⁸ + 21p⁴ + 21p⁴ + 9
Taking as the common term from the first two terms and 3 as the common term from the last two terms,
= 7p⁴(7p⁴ + 3) + 3(7p⁴ + 3)
Take (7p⁴+3) as the common term from the two terms,
= (7p⁴ + 3)(7p⁴ + 3)
Using the law of exponents, aⁿ × aˣ = a⁽ⁿ⁺ˣ⁾
= (7p⁴ + 3)⁽¹⁺¹⁾
= (7p⁴+3)²
Hence, the complete factorization of the polynomial 49p⁸ + 42p⁴ + 9 is (7p⁴+3)².
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