x^3+4x^2-2x-3 ; x+1 factor theorum to decide whether or not second polynomial is a factor of the first.
evaluate f(x) at x = -1
f(-1) = -1+4+2-3 = 2
it's not zero, so (x+1) is not a factor of f(x)
To determine whether the second polynomial, x + 1, is a factor of the first polynomial, x^3 + 4x^2 - 2x - 3, we can use the factor theorem.
The factor theorem states that if a polynomial f(x) is divided by (x - a), and the remainder is 0, then (x - a) is a factor of f(x).
In this case, we want to see if x + 1 is a factor of x^3 + 4x^2 - 2x - 3. We can use synthetic division to test this.
Let's set up synthetic division with the coefficients of the polynomial and the value -1 (since we are checking if x + 1 is a factor):
-1 | 1 4 -2 -3
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-1 -3 5
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The result of synthetic division is the bottom row of numbers: -1, -3, and 5.
If the remainder is 0, then x + 1 is a factor. In this case, the remainder is 5, which is not zero. Therefore, x + 1 is not a factor of x^3 + 4x^2 - 2x - 3.
Alternatively, if the remainder is zero, we can conclude that x + 1 is a factor of the polynomial.