5.4 Roots of Polynomials
This section is remixed from, Precalculus: An Investigation of Functions by Lippman and Rasmussen, licensed by Creative Commons 4.0 CC-BY-SA
In the last section, we limited ourselves to finding the intercepts, or zeros, of polynomials that factored simply, or we turned to technology. In this section, we will look at algebraic techniques for finding the zeros of polynomials like .
Long Division
In the last section we saw that we could write a polynomial as a product of factors, each corresponding to a horizontal intercept. If we knew that x = 2 was an intercept of the polynomial , we might guess that the polynomial could be factored as . To find that “something,” we can use polynomial division.
Examples of Polynomial Long Division
Divide by
Start by writing the problem out in long division form
Now we divide the leading terms: . It is best to align it above the same-powered term in the dividend. Now, multiply that by and write the result below the dividend.
Now subtract that expression from the dividend.
Again, divide the leading term of the remainder by the leading term of the divisor, . We add this to the result, multiply 6x by x-2, and subtract.
Repeat the process one last time.
This tells us divided by is , with a remainder of zero. This also means that we can factor as .
This gives us a way to find the intercepts of this polynomial. cannot be factored, but we could use the quadratic formula to find the roots.
. So all three roots are and
Try it Now 1
Divide by using long division.
The Factor and Remainder Theorems
When we divide a polynomial, p(x) by some divisor polynomial d(x), we will get a quotient polynomial q(x) and possibly a remainder r(x). In other words, .
Because of the division, the remainder will either be zero, or a polynomial of lower degree than d(x). Because of this, if we divide a polynomial by a term of the form , then the remainder will be zero or a constant.
If , then , which establishes the Remainder Theorem.
The Remainder Theorem
If is a polynomial of degree 1 or greater and c is a real number, then when p(x) is divided by , the remainder is .
If is a factor of the polynomial p, then for some polynomial q. Then , showing c is a zero of the polynomial. This shouldn’t surprise us – we already knew that if the polynomial factors, it reveals the roots.
If , then the remainder theorem tells us that if p is divided by , then the remainder will be zero, which means is a factor of p.
The Factor Theorem
If is a nonzero polynomial, then the real number c is a zero of if and only if is a factor of .
Synthetic Division
Since dividing by is a way to check if a number is a zero of the polynomial, it would be nice to have a faster way to divide by than having to use long division every time. Happily, quicker ways have been discovered. For more details on how synthetic division works, please see the source material. Recall the first example, we had:
Let’s re-work our division problem using a synthetic division tableau to see how it greatly streamlines the division process. To divide by , we write 2 in the place of the divisor and the coefficients of in for the dividend. Then “bring down” the first coefficient of the dividend.
Next, take the 2 from the divisor and multiply by the 1 that was “brought down” to get 2. Write this underneath the 4, then add to get 6.
Now take the 2 from the divisor times the 6 to get 12, and add it to the −5 to get 7.
Finally, take the 2 in the divisor times the 7 to get 14, and add it to the −14 to get 0.
The first three numbers in the last row of our tableau are the coefficients of the quotient polynomial. Remember, we started with a third degree polynomial and divided by a first degree polynomial, so the quotient is a second degree polynomial. Hence the quotient is . The last number is the remainder. Synthetic division is our tool of choice for dividing polynomials by divisors of the form x − c. It is important to note that it works only for these kinds of divisors. Also take note that when a polynomial (of degree at least 1) is divided by x − c, the result will be a polynomial of exactly one less degree. Finally, it is worth the time to trace each step in synthetic division back to its corresponding step in long division.
Examples Using Synthetic Division
a. Use synthetic division to divide by .
When setting up the synthetic division tableau, we need to enter 0 for the coefficient of x in the dividend. Doing so gives:
Since the dividend was a third degree polynomial, the quotient is a quadratic polynomial with coefficients 5, 13 and 39. Our quotient is and the remainder is r(x) = 118. This means
We can also write the remainder as a fraction over the divisor:
It also means that is not a factor of .
b. Divide by
For this division, we rewrite as and proceed as before.
The quotient is and the remainder is zero. Since the remainder is zero, is a factor of .
Try it Now 2
Divide by using synthetic division.
Using this process allows us to find the real zeros of polynomials, presuming we can figure out at least one root.
Example Finding Roots, given 1
The polynomial has a horizontal intercept at with multiplicity 2. Find the other intercepts of p(x).
Since is an intercept with multiplicity 2, then is a factor twice. Use synthetic division to divide by twice.
From the first division, we get
From the second division, we get
To find the remaining intercepts, we set
and get .
Note this also means
.
We could also think of this as
.
Try it Now Answers
1.
We can write this division as:
2.
divided by is: