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Question 1: Use grouping symbols to factorise the following:
A. [tex]x^3-2 x^2-16 x+32[/tex]
Answer (1)
Group the terms: ( x 3 − 2 x 2 ) + ( − 16 x + 32 ) .
Factor out common factors: x 2 ( x − 2 ) − 16 ( x − 2 ) .
Factor out the common binomial: ( x − 2 ) ( x 2 − 16 ) .
Factor the difference of squares: ( x − 2 ) ( x − 4 ) ( x + 4 ) . The final answer is ( x − 2 ) ( x − 4 ) ( x + 4 ) .
Explanation
Understanding the Problem We are asked to factorise the expression x 3 − 2 x 2 − 16 x + 32 using the grouping method. This involves grouping terms, factoring out common factors, and then factoring out a common binomial factor.
Grouping the Terms First, group the terms: ( x 3 − 2 x 2 ) + ( − 16 x + 32 ) .
Factoring out Common Factors Next, factor out the common factors from each group: x 2 ( x − 2 ) − 16 ( x − 2 ) .
Factoring out the Common Binomial Now, factor out the common binomial factor ( x − 2 ) : ( x − 2 ) ( x 2 − 16 ) .
Factoring the Difference of Squares Finally, factor the difference of squares x 2 − 16 = ( x − 4 ) ( x + 4 ) . Thus, the fully factorised expression is ( x − 2 ) ( x − 4 ) ( x + 4 ) .
Final Answer The factorised form of the expression x 3 − 2 x 2 − 16 x + 32 is ( x − 2 ) ( x − 4 ) ( x + 4 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures, ensuring stability and efficiency. Similarly, economists use factoring to analyze economic models and predict market trends. In computer graphics, factoring can help optimize rendering algorithms, making images load faster and smoother. Factoring is also used in cryptography to create secure codes and protect sensitive information.
