# Root and factorization of a polynomial

## Concept of root

The root or zero of a polynomial $$p(x)$$ is that value $$a$$ that $$p(a)=0$$$Mathematicians, throughout history, have always been fascinated by finding the roots of any polynomial. In general, this is a very complicated problem. So, using the remainder theorem and the factor theorem, we can deduce some properties of the roots of a polynomial: 1) The roots of a polynomial are divisors of the independent term. If it does not have an independent term, it means that it is divisible by $$x-a$$, where $$a=0$$, this is, it is divisible by $$x$$. $$p(x)=x^5+2x^4-3x^3+x^2-1$$ has as a root $$1$$, $$p(1)=1^5+2\cdot1^4-3\cdot1^3+1^2-1=0$$$ and $$1$$ divides the independent term $$-1$$.

The polynomial $$p(x)=2x^5+5x^4+4x^3-x^2+x$$ has the independent term equal to $$0$$.

Then, using the factor theorem, $$0$$ is a root of $$p(x)$$ and therefore $$x-0=x$$ divides the polynomial $$p(x)$$ exactly.

2) Being $$a_i$$ the $$i$$ roots of a polynomial, we can express this polynomial as product of polynomials like $$x-a_i$$.

The polynomial $$p(x)=x^2-3x+2$$ has roots $$x=2$$ and $$x=1$$. Therefore, it can be expressed as $$p(x)=(x-2)\cdot(x-1)$$$The polynomial $$p(x)=x^2+5x+6$$ has root $$x=-2$$ and $$x=-3$$. Therefore, it can be expressed as $$p(x)=(x+2)\cdot(x+3)$$$

3) A polynomial is called irreducible or prime if it does not have any rational number that is a root.

The polynomials $$p(x)=x^2+x+1$$ and $$q(x)=x^2+1$$ do not have any root in the rational numbers.

## Factorization of a polynomial

The process of factorization of a polynomial consists in finding all of its roots.

There are different techniques used to find the roots of a polynomial. Next, we will explain the most outstanding ones:

### Using notable products

The idea is to use the notable products but in the opposite sense. For example, if we know that: $$(a-b)\cdot(a+b)=a^2-b^2$$$It is clear that we can apply it the other way around: $$a^2-b^2=(a-b)\cdot(a+b)$$$ Therefore, if we have a polynomial like the following one $$x^2-16$$$which is $$x^2-4^2$$$ Applying the formula $$x^2-16=x^2-4^2=(x-4)(x+4)$$$This is, $$4$$ and $$-4$$ are the roots of the polynomial $$x^2-16$$. Factorize the following polynomial $$x^2-6x+9$$. We can see that the previous polynomial is a square of a difference: $$x^2-6x+9=x^2-2\cdot3\cdot x+3^2=(x-3)^2$$$ Therefore, the polynomial has the root $$x=3$$.

Factorize the following polynomial $$x^3+12x^2+48x+64$$.

We can see that the previous polynomial corresponds to the cube of a sum: $$x^3+12x^2+48x+64=x^3+3\cdot4\cdot x^2+3\cdot4^2x+4^3=(x+4)^3$$$Therefore, the polynomial has $$x=-4$$ as a root. ### Using the formula to solve quadratic equations If we have a polynomial $$p(x)$$ of degree $$2$$, we can equal it to $$0$$ and find the solution of the quadratic equation $$p(x)=0$$. These values solution will be the roots of the polynomial $$p(x)$$. Factorize the following polynomial $$x^2-x-2$$. We must solve the following equation $$x^2-x-2=0$$. We apply the formula to find the roots of a quadratic equation $$x=\dfrac{ -(-1)\pm\sqrt{ (-1)^2-4\cdot1\cdot(-2) } }{2\cdot1}=\dfrac{1\pm\sqrt{9}}{2}= \left\{\begin{array}{c} \dfrac{1+3}{2}=2 \\\\ \dfrac{1-3}{2}=-1 \end{array} \right.$$$

Therefore, the polynomial has $$x=2$$ and $$x=-1$$ as roots.

Factorize the following polynomial $$x^2+x-6$$.

We must solve the following equation $$x^2+x-6=0$$.

We apply the formula to find the roots of a quadratic equation $$x=\dfrac{ -1\pm\sqrt{ 1^2-4\cdot1\cdot(-6) } }{2\cdot1}=\dfrac{1\pm\sqrt{25}}{2}= \left\{\begin{array}{c} \dfrac{1+5}{2}=3 \\\\ \dfrac{1-5}{2}=-2 \end{array} \right.$$$Therefore, the polynomial has $$x=3$$ and $$x=-2$$ as its roots. ### Using the formula to solve biquadratic equations If we have a polynomial $$p(x)$$ of degree $$4$$ and even exponents, we can equal it to $$0$$ and find the solution of the biquadratic equation $$p(x)=0$$. These value solution will be the roots of the polynomial $$p(x)$$. Factorize the following polynomial $$x^4-5x^2+4$$. We must solve the following equation $$x^4-5x^2+4=0$$. We change the variable $$x^2=t$$ $$t^2-5t+4=0$$$

We apply the formula to find the roots of a quadratic equation $$x=\dfrac{ -(-5)\pm\sqrt{ (-5)^2-4\cdot1\cdot4 } }{2\cdot1}=\dfrac{5\pm\sqrt{9}}{2}= \left\{\begin{array}{c} \dfrac{5+3}{2}=4 \\\\ \dfrac{5-3}{2}=1 \end{array} \right.$$$Now we undo the change: $$x^2=4 \Rightarrow \left\{\begin{array}{c} x=2 \\\\ x=-2 \end{array} \right.$$$ $$x^2=1 \Rightarrow \left\{\begin{array}{c} x=1 \\\\ x=-1 \end{array} \right.$$$Therefore, the polynomial has $$x=2, x=-2, x=1$$ and $$x=-1$$ as its roots. Factorize the following polynomial $$x^4-10x^2+9$$. We must solve the following equation $$x^4-10x^2+9=0$$. We change the variable $$x^2=t$$ $$t^2-10t+9=0$$$

We apply the formula to find the roots of a quadratic equation $$x=\dfrac{ -(-10)\pm\sqrt{ (-10)^2-4\cdot1\cdot9 } }{2\cdot1}=\dfrac{10\pm\sqrt{64}}{2}= \left\{\begin{array}{c} \dfrac{10+8}{2}=9 \\\\ \dfrac{10-8}{2}=1 \end{array} \right.$$$Now we undo the change: $$x^2=9 \Rightarrow \left\{\begin{array}{c} x=3 \\\\ x=-3 \end{array} \right.$$$ $$x^2=1 \Rightarrow \left\{\begin{array}{c} x=1 \\\\ x=-1 \end{array} \right.$$$Therefore, the polynomial has $$x=3, x=-3, x=1$$ and $$x=-1$$ as its roots. ### Using the factor theorem For polynomials of a larger degree, our only tool is to use the factor theorem. This way, to find the roots of a polynomial, it will only be necessary to evaluate the polynomial for the values of $$x$$ that are divisors of the independent term, and for the values in which the expression turns out to be empty, these values will be the roots of the polynomial. With a few examples, we will visualize the procedure: Factorize the following polynomial $$p(x)=x^2-3x+2$$. As the properties of the factor theorem show, if $$a$$ is a root of $$p(x)$$, then $$p(a)=0$$. But, what value does $$a$$ have? There is a property that will be extremely useful: • If $$a$$ is a root of $$p(x)$$, $$a$$ is a divisor of the independent term of $$p(x)$$. In our case, the divisors of the independent term of the polynomial (of value $$2$$) are $$1,-1,2,-2$$$ Therefore, it is only necessary to evaluate these values in the polynomial and apply the factor theorem:

$$p(1)=1^2-3\cdot1+2=0$$

$$p(-1)=(-1)^2-3\cdot(-1)+2=6$$

$$p(2)=2^2-3\cdot2+2=0$$

$$p(-2)=(-2)^2-3\cdot(-2)+2=12$$

And so, the roots of $$p(x)$$ are $$x=1$$ and $$x=2$$.

Factorize the following polynomial $$p(x)=x^2+6x-7$$.

The divisors of the independent term of the polynomial (of value $$7$$) are: $$1,-1,7,-7$$\$ Therefore, it is only necessary to evaluate these values in the polynomial and apply the factor theorem:

$$p(1)=1^2+6\cdot1-7=0$$

$$p(-1)=(-1)^2+6\cdot(-1)-7=-12$$

$$p(7)=7^2+6\cdot7-7=84$$

$$p(-7)=(-7)^2+6\cdot(-7)-7=0$$

And so, the roots of $$p(x)$$ are $$x=1$$ and $$x=-7$$.