1. ## Remainder Theorem

Use the remainder theorem to factorise :

1) x^3 +3x^2 - 4

2) x^4 + 3x^3 + 5x^2 + 9x + 6

2. Originally Posted by Jen1603
Use the remainder theorem to factorise :

1) $f(x)=x^3 +3x^2 - 4$
The rational roots theorem tells us if this has rational roots, they will be
$\pm 1, \pm 2, \pm4$

Test each one using the remainder theorem to see which one(s) satisfy $f(x)=0$

I found $f(-2)=0$, so $x+2$ is a factor.

To find the remaining factors, you could test the rest of the possible rational roots using the remainder theorem or use synthetic division with divisor -2to reduce the polynomal to a quadratic that you can factor.

Code:

-2 | 1  3   0  -4
-2  -2   4
------------
1  1  -2  0
The quadratic remaining after factoring $x+2$ is $x^2+x-2$

That factors into $(x+2)(x-1)$

$\boxed{f(x)=(x+2)(x+2)(x-1)}$

Originally Posted by Jen1603
2) $f(x)= x^4 + 3x^3 + 5x^2 + 9x + 6$
Test these factors $\pm 1, \pm 2, \pm 3, \pm 6$

There are no sign changes in this polynomial, so no need to check for positive zero because there aren't any (Descartes Rule of Signs)