Question

How do you factor $0 = {x^3} + 4{x^2} - 3x - 18$?

Answer 1

Hint: To solve this question, we will find the quotient polynomial. After that, we will solve the question by applying the division method. Synthetic division is a shortcut way of dividing polynomials. It gives the same results as the polynomial long division but is much faster as it involves only the coefficients of the dividend and divisor, on which we perform basic arithmetic operations. As a result, we obtain the coefficients of the quotient and the remainder.
Steps to divide a polynomial by the binomial:
Set up the synthetic division.
Bring down the leading coefficient to the bottom row.
Multiply c by the value just written on the bottom row.
Add the column created in step 3.
Repeat until done.
Write out the answer.

Complete step by step solution:
In the given question, the polynomial is given as below.
 $ 0 = {x^3} + 4{x^2} - 3x - 18$
First, let us factor $0 = {x^3} + 4{x^2} - 3x - 18$ using the rational roots test.
If a polynomial function has integer coefficients, then every rational zero will have the form $\dfrac{p}{q}$ where p is a factor of the constant term and q is a factor of the leading coefficient.
Here, the constant term is 18. So, the value of p is equal to,
$p = \pm 1, \pm 18, \pm 2, \pm 9, \pm 3, \pm 6$
And the leading coefficient is 1.
So,
$q = \pm 1$
Find every combination of $ \pm \dfrac{p}{q}$. These are the possible roots of the polynomial function.
Substitute different values of p in the polynomial function.
$ \Rightarrow 0 = {x^3} + 4{x^2} - 3x - 18$
Let us take the value of p is equal to 2 in the polynomial function and simplify the expression.
$ \Rightarrow 0 = {\left( 2 \right)^3} + 4{\left( 2 \right)^2} - 3\left( 2 \right) - 18$
That is equal to,
$ \Rightarrow 0 = 8 + 4\left( 4 \right) - 6 - 18$
So,
$ \Rightarrow 0 = 8 + 16 - 6 - 18$
Let us simplify the right-hand side.
$ \Rightarrow 0 = 0$
Since 2 is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
$ \Rightarrow \dfrac{{{x^3} + 4{x^2} - 3x - 18}}{{x - 2}}$
Now, let us apply the Synthetic division method to find the remaining roots.
To divide by $x + 2$, we will perform the synthetic division with $x = - 2$

	${x^3}$	$4{x^2}$	$ - 3{x^1}$	-18	Row 0
	1	4	-3	-18	Row 1
+	0	2	12	18	Row 2
$ \times \left( 2 \right)$	1	6	9	0	Row 3
	${x^2}$	${x^1}$	${x^0}$	Remainder	Row 4

Hence, the answer is ${x^2} + 6x + 9$.
But, ${x^2} + 6x + 9$ is ${\left( {x + 3} \right)^2}$.

Hence, the factors of $0 = {x^3} + 4{x^2} - 3x - 18$ are $\left( {x - 2} \right){\left( {x + 3} \right)^2}$

Note:
Keep in mind that the division algorithm is:
Dividend=divisor (quotient) + remainder
$ \Rightarrow RHS = divisor\left( {quotient} \right) + remainder$