what is the factor of a^3 - 2a + 4?
please help me guys. :(
Hint:
A cubic has to have at least one zero.
If the expression has rational factors, there would be at least one linear factor fininishing in 1,2,4 or -1,-2,-4, since the coefficient of a^3 is 1.
For f(a)=a^3 - 2a + 4
So check f(1),f(2),... and which factor is in f(a). Do a long division and try to factor the remaining quadratic.
Sorry, the last paragraph is not clear.
If f(a0)=0, then (x-a0) is a factor of f(a). So check and see which of f(1),f(2)...=0 and hence find a rational factor of f(a).
If f(-1)=0, then (a+1) is a factor.
Do a long division and factor the remaining quadratic.
To find the factors of a polynomial, we can use polynomial factorization techniques. In this case, the given polynomial is a cubic polynomial, so we can try using either polynomial division or factoring by grouping.
1. Polynomial division:
To use polynomial division, we need to divide the polynomial by a potential factor and check if the remainder is zero. Let's start by trying a = 1 as a potential factor:
(a^3 - 2a + 4) / (a - 1) = a^2 + a - 3
Since the remainder is not zero, a = 1 is not a factor.
Now, we can try other potential factors such as a = 2, -1, -2, etc. until we find a factor that gives a remainder of zero.
2. Factoring by grouping:
If polynomial division does not yield a factor, we can try factoring by grouping. This method involves grouping the terms in a specific way to find common factors and then factoring them out.
The given polynomial, a^3 - 2a + 4, does not have any obvious common factors. Hence, we can't factor it using factoring by grouping.
If the factorization is represented as (a - r)(p(a) + q(a)), where r is the root or factor and p(a) and q(a) are the other two factors, then the factorization of a^3 - 2a + 4 will typically involve complex numbers or irrational numbers. In this case, without further information, we cannot determine the exact factorization of a^3 - 2a + 4.
It is important to note that some polynomial functions may not have easily identifiable factors or may require the use of numerical or algebraic methods to determine their factorization.