What is the completely factored form of [tex]p^4-16[/tex]?
A. [tex](p-2)(p-2)(p+2)(p+2)[/tex]
B. [tex](p-2)(p-2)(p-2)(p-2)[/tex]
C. [tex](p-2)(p+2)(p^2+2 o+4)[/tex]
D. [tex](p-2)(p+2)(p^2+4)[/tex]
Answer (1)
Recognize p 4 − 16 as a difference of squares and factor it into ( p 2 − 4 ) ( p 2 + 4 ) .
Factor p 2 − 4 as a difference of squares into ( p − 2 ) ( p + 2 ) .
Note that p 2 + 4 cannot be factored further using real numbers.
The completely factored form is ( p − 2 ) ( p + 2 ) ( p 2 + 4 ) .
Explanation
Analyze the problem We are asked to find the completely factored form of the expression p 4 − 16 . This looks like a difference of squares, which we can factor.
First factorization First, we recognize that p 4 − 16 can be written as ( p 2 ) 2 − 4 2 . This is a difference of squares, which factors as ( p 2 − 4 ) ( p 2 + 4 ) .
Second factorization Next, we notice that p 2 − 4 is also a difference of squares, since p 2 − 4 = p 2 − 2 2 . This factors as ( p − 2 ) ( p + 2 ) .
Check for further factorization The term p 2 + 4 is a sum of squares. It cannot be factored further using real numbers.
Final factored form Therefore, the completely factored form of p 4 − 16 is ( p − 2 ) ( p + 2 ) ( p 2 + 4 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in calculus and other advanced mathematics courses. For example, factoring is used to find the roots of a polynomial, which can represent the equilibrium points in a physical system or the optimal solutions in an economic model. Imagine you are designing a suspension bridge and need to calculate the forces acting on it. The equation describing the bridge's stability might involve a polynomial expression. By factoring this polynomial, you can determine the critical points where the forces are balanced, ensuring the bridge's structural integrity.
