Which shows the four-term polynomial and factored form of [tex]x^2+6 x-27[/tex]?
[tex]x^2+3 x-9 x-27=(x+3)(x-9)[/tex]
[tex]x^2+6 x-3 x-27=(x+6)(x-3)[/tex]
[tex]x^2+9 x-3 x-27=(x+9)(x-3)[/tex]
[tex]x^2+3 x-6 x-27=(x+3)(x-6)[/tex]
Answer (2)
Factor the quadratic expression x 2 + 6 x − 27 by finding two numbers that multiply to -27 and add to 6, which are 9 and -3.
Rewrite the middle term using these numbers: x 2 + 9 x − 3 x − 27 .
Factor by grouping: ( x 2 + 9 x ) + ( − 3 x − 27 ) = x ( x + 9 ) − 3 ( x + 9 ) = ( x + 9 ) ( x − 3 ) .
The correct four-term polynomial and factored form is x 2 + 9 x − 3 x − 27 = ( x + 9 ) ( x − 3 ) .
Explanation
Understanding the Problem We are given the quadratic expression x 2 + 6 x − 27 and asked to find the correct four-term polynomial and its factored form from the provided options.
Finding the Factors To factor the quadratic expression x 2 + 6 x − 27 , we need to find two numbers that multiply to -27 and add up to 6. These numbers are 9 and -3.
Rewriting the Middle Term Now, we rewrite the middle term using these two numbers: x 2 + 9 x − 3 x − 27 .
Factoring by Grouping Next, we factor by grouping: ( x 2 + 9 x ) + ( − 3 x − 27 ) = x ( x + 9 ) − 3 ( x + 9 ) = ( x + 9 ) ( x − 3 ) .
Identifying the Correct Option Finally, we verify that the factored form matches one of the given options. The correct option is x 2 + 9 x − 3 x − 27 = ( x + 9 ) ( x − 3 ) .
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, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where you know the area and need to determine the dimensions. Factoring the quadratic equation representing the area can help you find the possible lengths and widths of the garden, allowing you to choose the most suitable design.
The quadratic expression x 2 + 6 x − 27 factors to x 2 + 9 x − 3 x − 27 = ( x + 9 ) ( x − 3 ) . Therefore, the correct answer is that the four-term polynomial form is this expression, with the respective factors. The chosen option is the one that represents this factored form correctly.
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