What does 2^3 - 2(2)^2 + 2 - 2 simplify to?nae99 said:Homework Statement
show that (x-2) is a factor of x^3 - 2x^2 + x - 2
Homework Equations
The Attempt at a Solution
f(2) = 2^3 - 2(2)^2 + 2 - 2
is that any good
mark44 said:what does 2^3 - 2(2)^2 + 2 - 2 simplify to?
Mark44 said:OK, that's better. Now, you have f(2) = 0, where apparently f(x) = x^3 - 2x^2 + x - 2. If f(a) = 0, what does that tell you about x - a being a factor of f(x)?
That x - 2 is a factor of x^3 - 2x^2 + x - 2.nae99 said:that it is a factor of the equation
Mark44 said:That x - 2 is a factor of x^3 - 2x^2 + x - 2.
Note that x^3 - 2x^2 + x - 2 is not an equation (there's no equal sign).
nae99 said:show that (x-2) is a factor of x^3 - 2x^2 + x - 2
When (x-2) is a factor of x^3 - 2x^2 + x - 2, it means that when x is substituted in for (x-2), the resulting expression will equal 0. In other words, (x-2) is a root or solution of the polynomial x^3 - 2x^2 + x - 2.
To confirm that (x-2) is a factor of x^3 - 2x^2 + x - 2, you can use the factor theorem. This theorem states that if f(a) = 0, then (x-a) is a factor of f(x). So, substitute x=2 into the polynomial and if the resulting expression equals 0, then (x-2) is a factor.
The first step in factoring x^3 - 2x^2 + x - 2 is to look for any common factors. In this case, there are no common factors. Then, you can try to factor by grouping or using the rational root theorem. In this case, the rational root theorem can be used to find the possible rational roots of the polynomial. After testing the possible roots, we find that (x-2) is a factor. Then, we can use polynomial long division or synthetic division to factor out (x-2) from the polynomial.
No, (x-2) cannot be factored further from x^3 - 2x^2 + x - 2 because it is already a linear factor. The resulting quotient after factoring out (x-2) is x^2 + x + 1, which cannot be factored any further using real numbers.
The significance of (x-2) being a factor of x^3 - 2x^2 + x - 2 is that it tells us that 2 is a root or solution of the polynomial. This means that when x=2, the polynomial will equal 0. This information can be useful in solving equations or graphing the polynomial.