21. $1- x + x ^2- x ^3=$
a) $(1+x)\left(1-x^2\right)$
b) $(1-x)\left(1+x^2\right)$
c) $(1-x)\left(1-x^2\right)$
d) $(1-x)\left(1+x^2\right)$
Answer (2)
Group the terms: ( 1 − x ) + ( x 2 − x 3 ) .
Factor out x 2 : ( 1 − x ) + x 2 ( 1 − x ) .
Factor out ( 1 − x ) : ( 1 − x ) ( 1 + x 2 ) .
The correct factorization is ( 1 − x ) ( 1 + x 2 ) .
Explanation
Problem Analysis We are given the polynomial expression 1 − x + x 2 − x 3 and asked to find an equivalent factored form from the provided options.
Factoring by Grouping We can factor the polynomial by grouping terms. First, group the first two terms and the last two terms: ( 1 − x ) + ( x 2 − x 3 ) Then, factor out x 2 from the second group: ( 1 − x ) + x 2 ( 1 − x ) Now, factor out the common factor ( 1 − x ) from the entire expression: ( 1 − x ) ( 1 + x 2 ) This is the factored form of the given polynomial.
Comparing with Options Now we compare our factored form ( 1 − x ) ( 1 + x 2 ) with the given options: a) ( 1 + x ) ( 1 − x 2 ) b) ( 1 − x ) ( 1 + x 2 ) c) ( 1 − x ) ( 1 − x 2 ) d) ( 1 − x ) ( 1 + x 2 ) We see that option b) matches our factored form.
Final Answer Therefore, the correct factorization of the polynomial 1 − x + x 2 − x 3 is ( 1 − x ) ( 1 + x 2 ) . The answer is option b).
( 1 − x ) ( 1 + x 2 )
Examples
Factoring polynomials is a fundamental skill in algebra and is used extensively in various fields such as engineering, physics, and computer science. For example, in engineering, factoring polynomials can help simplify complex equations that model physical systems, making them easier to analyze and solve. In computer graphics, factoring can be used to optimize rendering algorithms, improving performance and efficiency. Understanding how to factor polynomials allows us to break down complex problems into simpler, more manageable parts.
The polynomial expression 1 − x + x 2 − x 3 can be factored as ( 1 − x ) ( 1 + x 2 ) . This matches option b) from the given choices. Thus, the correct answer is option b).
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