The polynomial $p(x)=x^3+3 x^2-4$ has a known factor of $(x-1)$. Rewrite $p(x)$ as a product of linear factors.
$p(x) = \square$
Answer (2)
Recognize that ( x − 1 ) is a factor of p ( x ) .
Perform polynomial division to find the quadratic factor: x 2 + 4 x + 4 .
Factor the quadratic: ( x + 2 ) 2 .
Express p ( x ) as a product of linear factors: p ( x ) = ( x − 1 ) ( x + 2 ) 2 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = x 3 + 3 x 2 − 4 and told that ( x − 1 ) is a factor. Our goal is to rewrite p ( x ) as a product of linear factors. This means we want to express p ( x ) in the form ( x − r 1 ) ( x − r 2 ) ( x − r 3 ) , where r 1 , r 2 , r 3 are the roots of the polynomial.
Polynomial Division Since ( x − 1 ) is a factor of p ( x ) , we can perform polynomial division to find the other factor. We can use synthetic division or polynomial long division. Let's perform polynomial division:
Divide x 3 + 3 x 2 − 4 by ( x − 1 ) .
x 3 + 3 x 2 − 4 = ( x − 1 ) ( x 2 + 4 x + 4 )
Factoring the Quadratic Now we have p ( x ) = ( x − 1 ) ( x 2 + 4 x + 4 ) . We need to factor the quadratic x 2 + 4 x + 4 . We can recognize this as a perfect square trinomial:
x 2 + 4 x + 4 = ( x + 2 ) ( x + 2 ) = ( x + 2 ) 2
Final Factorization Therefore, we can write p ( x ) as a product of linear factors:
p ( x ) = ( x − 1 ) ( x + 2 ) ( x + 2 ) = ( x − 1 ) ( x + 2 ) 2
Examples
Polynomial factorization is used in many areas of mathematics and engineering. For example, in control systems, the characteristic equation of a system is a polynomial, and the roots of this polynomial determine the stability of the system. Factoring the polynomial allows engineers to determine these roots and design stable systems. Similarly, in signal processing, polynomial factorization is used in filter design and spectral analysis.
The polynomial p ( x ) = x 3 + 3 x 2 − 4 can be rewritten as a product of linear factors as p ( x ) = ( x − 1 ) ( x + 2 ) 2 . This includes finding the quadratic factor after confirming a known factor and then factoring that quadratic. The final result incorporates both the known factor and the factored quadratic expression.
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