Factor completely $5 x^2+20 x$.
A. Prime
B. $5(x^2+4 x)$
C. $x(5 x+20)$
D. $5 x(x+4)$
Answer (2)
Identify the greatest common factor (GCF) of the terms.
Factor out the GCF from the expression.
Rewrite the expression as a product of the GCF and the remaining terms.
The complete factorization is 5 x ( x + 4 ) .
Explanation
Understanding the Problem We are asked to factor the expression 5 x 2 + 20 x completely. This means we want to write it as a product of simpler expressions that cannot be factored any further.
Finding the Greatest Common Factor First, we look for the greatest common factor (GCF) of the terms 5 x 2 and 20 x . The factors of 5 x 2 are 5 , x , and x , while the factors of 20 x are 2 , 2 , 5 , and x . The greatest common factor is 5 x .
Factoring out the GCF Now, we factor out the GCF, 5 x , from both terms: 5 x 2 + 20 x = 5 x ( x ) + 5 x ( 4 ) = 5 x ( x + 4 )
Final Factorization The expression x + 4 cannot be factored further, so the complete factorization of 5 x 2 + 20 x is 5 x ( x + 4 ) .
Examples
Factoring is a fundamental skill in algebra, and it's used in many real-world applications. For example, suppose you want to find the dimensions of a rectangular garden with an area of 5 x 2 + 20 x square feet. By factoring the expression, you can determine that the dimensions could be 5 x feet by ( x + 4 ) feet. This helps in planning the layout and fencing of the garden.
To factor the expression 5 x 2 + 20 x completely, we first identify the GCF as 5 x and then factor it out, resulting in 5 x ( x + 4 ) . The correct choice is D. 5 x ( x + 4 ) .
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