A student factors $10 x^2+3 x-27$ to the following:
$(2 x-3)(5 x+9)$
Which statement about $(2 x-3)(5 x+9)$ is true?
A. The expression is equivalent, and it is completely factored.
B. The expression is equivalent, but it is not completely factored.
C. The expression is not equivalent, but it is completely factored.
D. The expression is not equivalent, and it is not completely factored.
Answer (2)
Expand the factored expression ( 2 x − 3 ) ( 5 x + 9 ) to get 10 x 2 + 3 x − 27 .
Compare the expanded expression with the original expression 10 x 2 + 3 x − 27 . They are the same, so the expression is equivalent.
Check if the factors ( 2 x − 3 ) and ( 5 x + 9 ) can be factored further. Since they are linear, they cannot be factored further, so the expression is completely factored.
The expression is equivalent, and it is completely factored. The expression is equivalent, and it is completely factored.
Explanation
Problem Analysis We are given the quadratic expression 10 x 2 + 3 x − 27 and a student's factorization of it as ( 2 x − 3 ) ( 5 x + 9 ) . We need to determine if the student's factorization is correct and completely factored.
Expanding the Factored Expression To check if the factorization is correct, we will expand the factored expression and compare it to the original quadratic expression. Expanding ( 2 x − 3 ) ( 5 x + 9 ) , we get: ( 2 x − 3 ) ( 5 x + 9 ) = 2 x ( 5 x ) + 2 x ( 9 ) − 3 ( 5 x ) − 3 ( 9 ) = 10 x 2 + 18 x − 15 x − 27 = 10 x 2 + 3 x − 27
Checking Equivalence Comparing the expanded expression 10 x 2 + 3 x − 27 with the original expression 10 x 2 + 3 x − 27 , we see that they are identical. Therefore, the factored expression is equivalent to the original expression.
Checking Complete Factorization Now, we need to check if the factored expression is completely factored. The factors are ( 2 x − 3 ) and ( 5 x + 9 ) . Both of these are linear expressions, which means they cannot be factored further. Thus, the factored expression is completely factored.
Conclusion Since the expression is equivalent and completely factored, the correct statement is: The expression is equivalent, and it is completely factored.
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in various real-world applications. For example, engineers use factoring to design structures and analyze stress, while economists use it to model supply and demand curves. Imagine you are designing a rectangular garden with an area represented by the quadratic expression 10 x 2 + 3 x − 27 . Factoring this expression allows you to determine the possible dimensions (length and width) of the garden in terms of x , which can help you optimize the layout and use of space.
The student's factorization of 10 x 2 + 3 x − 27 as ( 2 x − 3 ) ( 5 x + 9 ) is correct and can be verified by expanding it back to the original expression. Both factors are linear and cannot be factored further, confirming that the expression is completely factored. Therefore, the correct choice is A: The expression is equivalent, and it is completely factored.
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