Question

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In this unit, you’re factoring a variety of polynomials. But not all polynomials can be factored! Here are some examples: x squared minus 6 x minus 5 x squared plus 4 3 x squared plus 2 x plus 1 x cubed plus 2 x squared plus 2 x plus 3 We might say that these polynomials cannot be factored further, or that they are completely factored. This means the expressions cannot be rewritten as the product of two or more quantities that divide them exactly. In this discussion, you’ll work with your classmates to examine some more examples of this. In your first post, write a polynomial that cannot be factored further. Explain in detail how you know it can’t be factored further. Describe the factoring techniques that might work with this polynomial, and show why they don’t work. Please follow these guidelines for your discussion posts: Write at least 150–300 words. Make sure your posts are grammatically and mechanically correct. Address all parts of the prompt. Provide at least one example to support your response. (Example choices include an anecdote, statistic, and/or textual evidence.)

Answer 1

One polynomial that cannot be factored further is x^2 + 2. This is a quadratic polynomial in which the coefficient of x^2 is 1, and the constant term is 2. To check if this polynomial can be factored further, we can use the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic polynomial (ax^2 + bx + c). In this case, a = 1, b = 0, and c = 2. Substituting these values into the formula, we get:

x = (-0 ± sqrt(0^2 - 4(1)(2))) / 2(1)
x = ±sqrt(-8)/2

Since the square root of a negative number is not a real number, we can conclude that x^2 + 2 cannot be factored further using real numbers. This is because the solutions to the quadratic equation are imaginary.

As for factoring techniques that might work with this polynomial, we can try factoring by grouping or factoring using the difference of squares formula. However, both of these techniques do not work for this polynomial since it cannot be expressed as the product of two or more quantities that divide it exactly.

In conclusion, x^2 + 2 is a polynomial that cannot be factored further using real numbers, and this is because the solutions to the quadratic equation are imaginary. Factoring by grouping and factoring using the difference of squares formula do not work for this polynomial since it cannot be expressed as the product of two or more quantities that divide it exactly.