Simplify: $\left(\frac{2 x^2+5 x+2}{x+1}\right)\left(\frac{x^2-1}{x+2}\right)$
A. $2 x^2-x-1$
B. $2 x^2-x-6$
C. $2 x^2+3 x+1$
D. $2 x^2-3 x+1$
Answer (2)
• Factorize the quadratic expression 2 x 2 + 5 x + 2 to get ( 2 x + 1 ) ( x + 2 ) .
• Factorize the quadratic expression x 2 − 1 to get ( x − 1 ) ( x + 1 ) .
• Rewrite the expression and cancel out the common terms ( x + 1 ) and ( x + 2 ) .
• Simplify the expression to get 2 x 2 − x − 1 , so the answer is 2 x 2 − x − 1 .
Explanation
Understanding the Problem We are given the expression ( x + 1 2 x 2 + 5 x + 2 ) ( x + 2 x 2 − 1 ) to simplify. Our goal is to factorize the quadratic expressions, cancel out common terms, and simplify the expression to match one of the given options.
Factorizing the First Quadratic Expression First, let's factorize the quadratic expression 2 x 2 + 5 x + 2 . We are looking for two numbers that multiply to 2 × 2 = 4 and add up to 5 . These numbers are 4 and 1 . So we can rewrite the expression as:
2 x 2 + 4 x + x + 2 = 2 x ( x + 2 ) + 1 ( x + 2 ) = ( 2 x + 1 ) ( x + 2 )
Factorizing the Second Quadratic Expression Next, let's factorize the quadratic expression x 2 − 1 . This is a difference of squares, so we have:
x 2 − 1 = ( x − 1 ) ( x + 1 )
Rewriting the Expression Now, we can rewrite the original expression with the factorized forms:
( x + 1 ( 2 x + 1 ) ( x + 2 ) ) ( x + 2 ( x − 1 ) ( x + 1 ) )
Cancelling Common Terms We can now cancel out the common terms ( x + 1 ) and ( x + 2 ) in the numerator and the denominator:
( x + 1 ) ( x + 2 ) ( 2 x + 1 ) ( x + 2 ) ( x − 1 ) ( x + 1 ) = ( 2 x + 1 ) ( x − 1 )
Simplifying the Expression Finally, let's simplify the expression by expanding the product:
( 2 x + 1 ) ( x − 1 ) = 2 x 2 − 2 x + x − 1 = 2 x 2 − x − 1
So, the simplified expression is 2 x 2 − x − 1 .
Choosing the Correct Option Comparing the simplified expression 2 x 2 − x − 1 with the given options, we see that it matches option A.
Therefore, the correct answer is A.
Examples
Simplifying algebraic expressions is a fundamental skill in many areas of mathematics and science. For example, in physics, you might need to simplify an expression for the kinetic energy of an object or the potential energy of a system. In engineering, you might need to simplify expressions for the stress or strain in a material. By simplifying these expressions, you can make them easier to work with and gain a better understanding of the underlying relationships.
The expression ( x + 1 2 x 2 + 5 x + 2 ) ( x + 2 x 2 − 1 ) simplifies to 2 x 2 − x − 1 , which corresponds to option A.
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