Which expression is the factorization of [tex]x^2+10 x+21[/tex]?
[tex](x+3)(x+7)[/tex]
[tex](x+4)(x+6)[/tex]
[tex](x+6)(x+15)[/tex]
[tex](x+7)(x+14)[/tex]
Answer (2)
The factorization of the expression x 2 + 10 x + 21 is ( x + 3 ) ( x + 7 ) . This is confirmed by checking that the numbers 3 and 7 multiply to 21 and add to 10. Therefore, the correct answer is ( x + 3 ) ( x + 7 ) .
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We are given the quadratic expression x 2 + 10 x + 21 and need to find its factorization.
Expand the first option ( x + 3 ) ( x + 7 ) : ( x + 3 ) ( x + 7 ) = x 2 + 7 x + 3 x + 21 = x 2 + 10 x + 21 .
Since the expansion matches the original expression, ( x + 3 ) ( x + 7 ) is the correct factorization.
The correct factorization is ( x + 3 ) ( x + 7 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 10 x + 21 and asked to find its factorization from the given options. The correct factorization should expand to the original quadratic expression.
Checking the First Option Let's examine the first option, ( x + 3 ) ( x + 7 ) . Expanding this gives us: ( x + 3 ) ( x + 7 ) = x 2 + 7 x + 3 x + 21 = x 2 + 10 x + 21
Confirming the Answer Since ( x + 3 ) ( x + 7 ) expands to x 2 + 10 x + 21 , it is the correct factorization. We can also check the other options to confirm they are incorrect.
Checking the Second Option The second option is ( x + 4 ) ( x + 6 ) . Expanding this gives us: ( x + 4 ) ( x + 6 ) = x 2 + 6 x + 4 x + 24 = x 2 + 10 x + 24 This is not equal to x 2 + 10 x + 21 .
Checking the Third Option The third option is ( x + 6 ) ( x + 15 ) . Expanding this gives us: ( x + 6 ) ( x + 15 ) = x 2 + 15 x + 6 x + 90 = x 2 + 21 x + 90 This is not equal to x 2 + 10 x + 21 .
Checking the Fourth Option The fourth option is ( x + 7 ) ( x + 14 ) . Expanding this gives us: ( x + 7 ) ( x + 14 ) = x 2 + 14 x + 7 x + 98 = x 2 + 21 x + 98 This is not equal to x 2 + 10 x + 21 .
Final Answer Therefore, the correct factorization of x 2 + 10 x + 21 is ( x + 3 ) ( x + 7 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and calculate stresses, while economists use it to model supply and demand curves. Imagine you are designing a rectangular garden with an area represented by the expression x 2 + 10 x + 21 . By factoring this expression into ( x + 3 ) ( x + 7 ) , you determine that the dimensions of the garden are ( x + 3 ) and ( x + 7 ) . This allows you to plan the layout and fencing requirements efficiently.
