f(x) = x3 + 2x2 – 0.4X – 1
how do you you factor this?
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http://www.jiskha.com/display.cgi?id=1276043019
Please use ^ before exponents.
There is no simpl standard formula for finding the roots of cubic equations. First create integer coefficients by rewriting as
(1/5) (5x^3 + 10x^2 -2x -5)
There are no simple factors with integers for this equation. That can be proven with something sometimes called the "p/q theorem". It can be faotored to
f(x) = 5[(x-0.689)(x+1.941,(x+0.748)]
I used an online solver for that.
The numbers in the factors are rounded.
To factor the given polynomial f(x) = x³ + 2x² – 0.4x – 1, you can follow these steps:
Step 1: Check for common factors
Look for any common factors in all the terms of the polynomial. In this case, there are no common factors other than 1.
Step 2: Find the possible rational roots
Use the Rational Root Theorem to determine the possible rational roots of the polynomial. The Rational Root Theorem states that if a rational number r is a root of a polynomial equation with integer coefficients, then r must be a factor of the constant term divided by the leading coefficient. In this case, the constant term is -1 and the leading coefficient is 1, so the possible rational roots are ±1.
Step 3: Use synthetic division or long division
You can now use synthetic division or long division to test these possible rational roots. Let's start by testing 1 as a root:
Performing synthetic division with 1 as a root:
```
1 | 1 2 -0.4 -1
| 1 3 2.6
─────────────────
1 3 2.6 1.6
```
The result of the synthetic division is 1x² + 3x + 2.6 with a remainder of 1.6.
Since the remainder is not zero, 1 is not a root of the polynomial. Now, let's test -1 as a root:
Performing synthetic division with -1 as a root:
```
-1 | 1 2 -0.4 -1
| -1 -1 1.4
─────────────────
1 1 -1.4 0.4
```
The result of the synthetic division is 1x² + x - 1.4 with a remainder of 0.4.
Since the remainder is not zero, -1 is also not a root of the polynomial.
Step 4: Factor the polynomial
Since the possible rational roots did not yield a zero remainder, the polynomial cannot be factored with rational numbers. Therefore, the polynomial f(x) = x³ + 2x² – 0.4x – 1 is not easily factorable using integer or rational numbers.
In such cases, you may need to resort to numerical methods, such as using graphing calculators or software, to find the approximate solutions or use more advanced algebraic techniques.