Simplify the expression below.
$\frac{12 x^3-14 x^2-40 x}{2 x-5}$
A. $2 x^3+3 x$
B. $3 x^2-15$
C. $6 x^2+8 x$
D. $7 x^2-4 x$
Answer (2)
Factor 2 x from the numerator: 12 x 3 − 14 x 2 − 40 x = 2 x ( 6 x 2 − 7 x − 20 ) .
Factor the quadratic expression: 6 x 2 − 7 x − 20 = ( 3 x + 4 ) ( 2 x − 5 ) .
Substitute the factored form into the expression: 2 x − 5 2 x ( 3 x + 4 ) ( 2 x − 5 ) .
Cancel the common factor ( 2 x − 5 ) and simplify: 2 x ( 3 x + 4 ) = 6 x 2 + 8 x . The simplified expression is 6 x 2 + 8 x .
Explanation
Problem Analysis We are asked to simplify the rational expression 2 x − 5 12 x 3 − 14 x 2 − 40 x .
Factoring the Numerator First, we factor out the common factor 2 x from the numerator: 12 x 3 − 14 x 2 − 40 x = 2 x ( 6 x 2 − 7 x − 20 ) . So the expression becomes 2 x − 5 2 x ( 6 x 2 − 7 x − 20 ) .
Factoring the Quadratic Expression Now we need to factor the quadratic expression 6 x 2 − 7 x − 20 . We look for two numbers that multiply to 6 × − 20 = − 120 and add up to − 7 . These numbers are − 15 and 8 . So we can rewrite the quadratic as 6 x 2 − 15 x + 8 x − 20 .
Factoring by Grouping Next, we factor by grouping: 6 x 2 − 15 x + 8 x − 20 = 3 x ( 2 x − 5 ) + 4 ( 2 x − 5 ) = ( 3 x + 4 ) ( 2 x − 5 ) .
Substituting Back Substitute the factored form back into the expression: 2 x − 5 2 x ( 3 x + 4 ) ( 2 x − 5 ) .
Canceling Common Factors Now we can cancel the common factor ( 2 x − 5 ) from the numerator and the denominator: 2 x − 5 2 x ( 3 x + 4 ) ( 2 x − 5 ) = 2 x ( 3 x + 4 ) .
Final Simplification Finally, we distribute 2 x to get: 2 x ( 3 x + 4 ) = 6 x 2 + 8 x . Therefore, the simplified expression is 6 x 2 + 8 x .
Examples
Simplifying rational expressions is a fundamental skill in algebra and calculus. For example, in physics, you might encounter a complex expression describing the motion of an object. Simplifying this expression can make it easier to analyze the object's trajectory or velocity. Similarly, in engineering, simplifying expressions can help optimize designs or predict the behavior of a system. This skill is also crucial in computer science for optimizing algorithms and reducing computational complexity. By mastering simplification techniques, you can tackle more complex problems in various fields, making your work more efficient and insightful.
The expression 2 x − 5 12 x 3 − 14 x 2 − 40 x simplifies to 6 x 2 + 8 x . Thus, the correct choice is C. 6 x 2 + 8 x .
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