Question

How do you factor \[{{x}^{3}}+5{{x}^{2}}-9x-45?\]

Answer 1

Hint: We are given an equation \[{{x}^{3}}+5{{x}^{2}}-9x-45\] and we are asked to factor it as much as possible. Now, we will first learn about the type of the equation then with the type of equation and number of terms, we will try methods to find the factor. We will use the method of grouping to find the factor. We will also use \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\] to find our required factored method.

Complete step-by-step solution:
We are given an equation as \[{{x}^{3}}+5{{x}^{2}}-9x-15.\] We can see that the highest power of our given equation is 3 so it is a three-degree polynomial. The number of degrees will tell us the possible number of factors as its degree is 3 so it can have a maximum of 3 factors. Now, as we can see that our equation has 4 terms, so we use the method of grouping. In this, we will pair the first two terms and pair the last two terms and then find the possible common factor.
Now, we have,
\[{{x}^{3}}+5{{x}^{2}}-9x-45={{x}^{2}}\left( x+5 \right)-9\left( x+5 \right)\]
[As \[{{x}^{2}}\] was common in the first two terms and – 9 is common in the last two terms]
Now, as we can see that (x + 5) is the same for both, so we get,
\[\Rightarrow \left( x+5 \right)\left[ {{x}^{2}}-9 \right]\]
So, we get,
\[\Rightarrow {{x}^{3}}+5{{x}^{2}}-9x-45=\left( x+5 \right)\left[ {{x}^{2}}-9 \right]\]
Now as \[9={{3}^{2}}\] so we get,
\[\Rightarrow {{x}^{3}}+5{{x}^{2}}-9x-45=\left( x+5 \right)\left( {{x}^{2}}-{{3}^{2}} \right)\]
Now, we will use \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\] to factor furthermore.
Consider x = a and 3 = b, we get,
\[\Rightarrow {{x}^{2}}-{{3}^{2}}=\left( x+3 \right)\left( x-3 \right)\]
Using this above, we get,
\[\Rightarrow {{x}^{3}}+5{{x}^{2}}-9x-45=\left( x+5 \right)\left( x+3 \right)\left( x-3 \right)\]
So, we get that factor form of \[{{x}^{3}}+5{{x}^{2}}-9x-45=\left( x+5 \right)\left( x+3 \right)\left( x-3 \right).\]

Note: While looking for a factor we always first look at the term and then work on its structure form, say if we have the term as \[{{x}^{3}}-{{y}^{3}}\] so we can use the expansion to find the factor of it. The degree of the polynomial always helps us to know where to stop as if we know the degree is 2. So, we will not find after we get 2 factors and the key thing is to always cross-check the term by solving so that it eliminates the chances of error as its long calculation mistake can happen anytime.