Factor $x^4+3 x^2-28$
Answer (1)
We factor the given polynomial x 4 + 3 x 2 − 28 by first substituting y = x 2 to obtain a quadratic y 2 + 3 y − 28 . Then we factor the quadratic as ( y + 7 ) ( y − 4 ) . Substituting x 2 back in for y , we get ( x 2 + 7 ) ( x 2 − 4 ) . Finally, we factor the difference of squares x 2 − 4 as ( x − 2 ) ( x + 2 ) . The complete factorization is ( x 2 + 7 ) ( x − 2 ) ( x + 2 ) .
Explanation
Understanding the Problem We are given the quartic polynomial x 4 + 3 x 2 − 28 and asked to factor it. We can rewrite this polynomial as a quadratic in x 2 and then factor that quadratic.
Substitution Let y = x 2 . Then the polynomial becomes y 2 + 3 y − 28 . We want to factor this quadratic expression.
Factoring the Quadratic We look for two numbers that multiply to − 28 and add to 3 . These numbers are 7 and − 4 . Thus, we can factor the quadratic as ( y + 7 ) ( y − 4 ) .
Substituting Back Now we substitute x 2 back in for y to get ( x 2 + 7 ) ( x 2 − 4 ) .
Factoring Difference of Squares We can further factor the term x 2 − 4 as a difference of squares: x 2 − 4 = ( x − 2 ) ( x + 2 ) .
Final Factorization Therefore, the complete factorization is ( x 2 + 7 ) ( x − 2 ) ( x + 2 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many areas of mathematics and engineering. For example, when designing a bridge, engineers need to analyze the forces acting on the bridge. This often involves solving polynomial equations, which requires factoring polynomials. Factoring also helps in simplifying complex expressions, making them easier to work with. Understanding how to factor polynomials can help you solve real-world problems more efficiently.
