f(x) = x3 + 2x2 – 0.4X – 1
how do you you factor this?
Is this supposed to be
f(x) = x^3 + 2x^2 - .4x - 1 ??
This does not factor, since
x^3 + 2x^2 - .4x - 1 = 0 does not have rational roots
To factor the given polynomial, f(x) = x^3 + 2x^2 - 0.4x - 1, we can follow these steps:
Step 1: Look for the greatest common factor (GCF) of all the terms in the polynomial. In this case, there is no common factor other than 1.
Step 2: Check for potential rational roots using the Rational Root Theorem. According to the Rational Root Theorem, the possible rational roots of the polynomial are all the divisors of the constant term (-1) divided by all the divisors of the leading coefficient (1). So, the possible rational roots are ±1/1, ±1/2, ±1/5, and ±1.
Step 3: Use synthetic division or long division to determine if any of the potential rational roots found in step 2 are actually roots of the polynomial. Synthetic division is a quicker method to test the potential roots.
Let's use synthetic division to determine if any of the potential rational roots are roots of the given polynomial.
Since there is no specific root to test, we will try one of the potential rational roots, let's start with x = 1.
1 | 1 2 -0.4 -1
| 1 3.3 2.9
----------------------
1 3.3 2.9
The result of synthetic division shows that the remainder is not zero when dividing by x - 1. This means that x = 1 is not a root of the given polynomial.
You can carry on testing other potential rational roots using the same synthetic division process. The rational root with zero remainder will indicate the presence of a root.
If you continue the synthetic division testing different rational roots, you will find that the polynomial does not have any rational roots. Hence, it cannot be factored further using rational numbers.
Therefore, the given polynomial f(x) = x^3 + 2x^2 - 0.4x - 1 cannot be factored using rational numbers.