f. $x^{10}-10 x^6+9 x^2$
Answer (1)
Factor out the common factor: x 10 − 10 x 6 + 9 x 2 = x 2 ( x 8 − 10 x 4 + 9 ) .
Substitute y = x 4 : x 8 − 10 x 4 + 9 = y 2 − 10 y + 9 .
Factor the quadratic: y 2 − 10 y + 9 = ( y − 1 ) ( y − 9 ) .
Substitute back and factor further: x 2 ( x − 1 ) ( x + 1 ) ( x 2 + 1 ) ( x − 3 ) ( x + 3 ) ( x 2 + 3 ) .
Explanation
Problem Analysis We are given the polynomial x 10 − 10 x 6 + 9 x 2 and we want to factor it completely.
Factoring out common factor First, we factor out the common factor x 2 from the polynomial: x 10 − 10 x 6 + 9 x 2 = x 2 ( x 8 − 10 x 4 + 9 ) .
Substitution Now, let y = x 4 . Then the polynomial inside the parenthesis becomes a quadratic expression in y : x 8 − 10 x 4 + 9 = y 2 − 10 y + 9 .
Factoring the quadratic We can factor the quadratic expression y 2 − 10 y + 9 as follows: y 2 − 10 y + 9 = ( y − 1 ) ( y − 9 ) .
Substituting back Now, substitute back x 4 for y : ( y − 1 ) ( y − 9 ) = ( x 4 − 1 ) ( x 4 − 9 ) .
Factoring difference of squares We can factor x 4 − 1 as a difference of squares: x 4 − 1 = ( x 2 − 1 ) ( x 2 + 1 ) = ( x − 1 ) ( x + 1 ) ( x 2 + 1 ) .
Factoring difference of squares Similarly, we can factor x 4 − 9 as a difference of squares: x 4 − 9 = ( x 2 − 3 ) ( x 2 + 3 ) = ( x − 3 ) ( x + 3 ) ( x 2 + 3 ) .
Complete Factorization Combining all the factors, we obtain the complete factorization: x 10 − 10 x 6 + 9 x 2 = x 2 ( x − 1 ) ( x + 1 ) ( x 2 + 1 ) ( x − 3 ) ( x + 3 ) ( x 2 + 3 ) .
Final Answer Therefore, the factored form of the given polynomial is x 2 ( x − 1 ) ( x + 1 ) ( x 2 + 1 ) ( x − 3 ) ( x + 3 ) ( x 2 + 3 ) .
Examples
Factoring polynomials is a fundamental concept in algebra and has many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model economic growth, and computer scientists use it to design efficient algorithms. In cryptography, factoring large numbers is crucial for the security of encryption methods. Understanding factoring helps in simplifying complex expressions and solving equations, making it an essential skill in various fields.
