Factor completely $x^3-2 x^2-5 x+10$
A. $(x-2)(x^2-5)$
B. $(x+2)(x^2+5)$
C. $(x-2)(x^2+5)$
D. $(x+2)(x^2-5)$
Answer (2)
Group the terms: ( x 3 − 2 x 2 ) + ( − 5 x + 10 ) .
Factor out the GCF from each group: x 2 ( x − 2 ) − 5 ( x − 2 ) .
Factor out the common binomial: ( x − 2 ) ( x 2 − 5 ) .
The completely factored form is: ( x − 2 ) ( x 2 − 5 ) .
Explanation
Problem Analysis We are given the polynomial x 3 − 2 x 2 − 5 x + 10 and asked to factor it completely. We can use factoring by grouping to achieve this.
Grouping Terms First, group the terms in pairs: ( x 3 − 2 x 2 ) + ( − 5 x + 10 ) .
Factoring out GCF Next, factor out the greatest common factor (GCF) from each pair. From the first pair, x 3 − 2 x 2 , we can factor out x 2 , which gives us x 2 ( x − 2 ) . From the second pair, − 5 x + 10 , we can factor out − 5 , which gives us − 5 ( x − 2 ) . So we have x 2 ( x − 2 ) − 5 ( x − 2 ) .
Factoring out Common Binomial Now, we can see that ( x − 2 ) is a common factor in both terms. Factoring out ( x − 2 ) gives us ( x − 2 ) ( x 2 − 5 ) .
Factoring Difference of Squares The expression x 2 − 5 can be further factored as a difference of squares. Since 5 = ( 5 ) 2 , we have x 2 − 5 = x 2 − ( 5 ) 2 = ( x − 5 ) ( x + 5 ) .
Final Factorization Therefore, the complete factorization of the given polynomial is ( x − 2 ) ( x − 5 ) ( x + 5 ) . However, the question only asks for factorization until ( x − 2 ) ( x 2 − 5 ) .
Final Answer Thus, the factored form of the polynomial is ( x − 2 ) ( x 2 − 5 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. Imagine you are designing a bridge, and the equation describing the load on the bridge is a polynomial. By factoring this polynomial, you can find the critical points where the load is highest, ensuring the bridge is strong enough to withstand the stress. Similarly, in physics, factoring can help simplify equations describing the motion of objects or the behavior of waves.
The polynomial x 3 − 2 x 2 − 5 x + 10 can be factored completely as ( x − 2 ) ( x 2 − 5 ) . This matches option A from the multiple-choice answers provided. Factoring involves grouping terms and extracting common factors to simplify the expression.
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