Factors of Polynomials, Part 2

So we are talking about polynomials and how to factor them. We want to factor a polynomial in order to easily find its zeroes, that is, the values of x that make the polynomial equal to 0.

First a definition: the order or degree of a polynomial, is the highest power of x in the polynomial. So x² -3x + 7 is a polynomial of degree 2.

The first method to discuss is the easiest to apply. If the polynomial has an x in each term, you can factor that out. This will show that x itself is a factor of the polynomial. Let’s do an example.

Consider

${x}^{3}\hspace{0.33em}{-}\hspace{0.33em}{11}{x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{30}{x}$

Notice that there is at least one x in each term. We can “undistribute” this x and make it a factor. Please review my posts on the Distributive Property.

${x}^{3}\hspace{0.33em}{-}\hspace{0.33em}{11}{x}^{2}\hspace{0.33em}{+}\hspace{0.33em}{30}{x}\hspace{0.33em}{=}\hspace{0.33em}{x}{(}{x}^{2}\hspace{0.33em}{-}\hspace{0.33em}{11}{x}\hspace{0.33em}{+}\hspace{0.33em}{30}{)}$

Since x by itself is a factor of the polynomial, 0 itself is a zero of the polynomial. So now we need to complete the factoring process by factoring the stuff in the brackets.

The expression left in the brackets is called a quadratic which means it’s a polynomial of degree 2. Now I’ve discussed methods before on how to factor or find the zeroes of a quadratic. I will review these in my next post.