(x-2) Is A Factor Of X^4+2x^3-7x^2-8x+12

Jasonbradley

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Jun 17, 2024

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Determining if (x-2) is a factor of x^4 + 2x^3 - 7x^2 - 8x + 12

In algebra, a factor of a polynomial is a polynomial that divides evenly into the original polynomial. We can use the Factor Theorem to determine if a binomial of the form (x-a) is a factor of a polynomial.

The Factor Theorem: A binomial (x-a) is a factor of a polynomial p(x) if and only if p(a) = 0.

Let's apply this to our problem:

1. Identify 'a': In our binomial (x-2), a = 2.

2. Substitute 'a' into the polynomial: We need to find the value of the polynomial when x = 2:

   • p(2) = (2)^4 + 2(2)^3 - 7(2)^2 - 8(2) + 12
   • p(2) = 16 + 16 - 28 - 16 + 12
   • p(2) = 0

3. Interpret the result: Since p(2) = 0, the Factor Theorem tells us that (x-2) is indeed a factor of the polynomial x^4 + 2x^3 - 7x^2 - 8x + 12.

Further Exploration:

• We can use polynomial long division or synthetic division to divide the polynomial by (x-2) and obtain the quotient, which will be another factor of the original polynomial.
• This process can be repeated to factor the polynomial completely.