Using The Rational Zeros Theorem

In this lesson, you will run across the rational root theorem, or the rational zero theorem, and how to use it with guided examples.

Later on that, there are test style practice questions for you!

If you're comfortable with what rational numbers and roots are, then you can skip alee to the theorem itself.

Keep this tabular array of factors and multiples handy while y'all're working through the examples and practice questions!

Contents

What is a Rational Number?
What is a Root?
The Rational Root Theorem (Rational Zero Theorem)
How to Use It
Examples
Why Utilize Information technology?
What is the Integral Root Theorem?
Practice
To Sum Up (Pun Intended!)

What is a Rational Number?

A ratio is a fractional relationship between two quantities!

A ratio-nal number is a quantity that can be written as a fraction.

What is a Root?

A root is a value for 10 that makes the function equal to zero. It is also called a solution.

This role f(x) is a polynomial of order 3. The highest ability of x is iii.

The graph of the function crosses the x-axis at points x=-ane and x=1. These are the x-intercepts because at these points on the office, y=0.

So the roots, or solutions, of function f(x) are x=-1 and x=1.

In the instance above, the root x=-1 is repeated because information technology is also a local maximum.

The Rational Root Theorem (Rational Zero Theorem)

Also known as the rational zero theorem, the rational root theorem is a powerful mathematical tool used to discover all possible rational roots of a polynomial equation of the order 3 and above.

The rational root theorem says that if in that location are rational roots, they will be i of the following:

This ways that the roots of the equation are ane of the combinations of ± the factors of the constant-coefficient a₀ divided past ± the factors of the n^coefficient a_n .

For the theorem to work, the coefficients a_n , a_n-1 , a_north-ii , etc., must be integers – whole numbers.

Yous'll see how information technology works in the next section.

Only use the rational root theorem when the coefficients are small, and every bit a last resort or when you're told to. Information technology is less efficient than some other steps y'all could take beginning.

Beginning, endeavor factoring every bit much as possible, it may turn out you don't need to utilise this method if you can simplify a polynomial of a college society, similar x^three or ten^four, past reducing it to a combination of lower-order equation(south), for instance:

In that case, there is no need to use a theorem to solve the equation – factoring did it for you!

How to Use the Rational Root Theorem

First, identify the coefficients you will need to utilise.

Let'due south say that you've been asked to list all the possible rational roots for the last equation in the diagram in a higher place:

threex³-4x²-17x+six=0

Utilise the rational root theorem.

The coefficients are a_north=three and a₀=six.

The factors of 3 are one and 3, and the factors of half dozen are 1, ii, 3, and 6. And then the possible roots are:

Put each number into the role equally x to test if it's a root, similar this.

If f(10)=0, and then the value is a rational root, so you can meet that putting x={-ii, ¹⁄₃, 3} gives 0.

And then the polynomial has 3 rational roots, x=-ii, x=ⁱ⁄₃, and x=3.

A polynomial always has the same number of roots as its order.

Reminder: a polynomial'due south order is the highest power of the variable – in most cases, the variable is ten or t.

So a polynomial of order 5 has 5 roots. That doesn't mean they are all rational though.

A polynomial can take no rational, or existent, roots. This just means that it'south roots are complex, and involve some irrational operator, like a radical, e, π, etc.

If the theorem finds no zeros, the polynomial has no rational roots.

Examples

a) List the possible rational roots for the function

f(ten) = x⁴ + 2x^three – 7x² – 8x + 12

b) Test each possible rational root in the function to confirm which are solutions to f(x)=0.

c) Use the confirmed rational roots to factorize the polynomial.

a) To notice the possible rational roots, use the theorem: ± the factors of the constant-coefficient 12 divided past the factors of the x⁴-coefficient 1.

b) For each possible rational root, replace ten with the value and evaluate the part.

c) The confirmed roots are the ones that made the function equal to zero.

In this terminal case, did you notice that when nosotros constitute roots, similar negative three, the 3 is positive when writing out the factorization? The sign changed. Why is that?

Information technology's because each subclass in the factorization must equal zero when 10 is that root. A root of -3 will make cipher when 3 is added to it.

At the point in the graph where ten=-three, (x+3) is actually (-3+3)=0.

The whole function multiplied by 0 is 0!

a) Listing the possible rational roots for the function

f(ten) = 2x^three + 7x² – 37x – 42

Hither are the factors of 42 for your reference.

b) Test each possible rational root in the role to confirm which are solutions to f(x)=0.

c) Employ the confirmed rational roots to factorize the polynomial.

a) To find the possible rational roots, utilise the theorem: ± the factors of the constant-coefficient, 42, divided past the factors of the x^three-coefficient, 2.

b) For each possible rational root, replace 10 with the value and evaluate the part.

c) The confirmed roots are the ones that made the function equal to aught.

Why Employ It?

In advanced mathematics, there will be times when you must observe the solutions or roots of a polynomial for a practical, or theoretical purpose.

You may want to find the indicate of greatest menses while on the blueprint team for a rollercoaster at a theme park, which can be modeled using a polynomial equation, or mayhap you'll need to theoretically test the aerodynamics of a curved border before building an expensive prototype.

What is the Integral Root Theorem?

At that place is a special example of the rational root theorem, where the coefficient a_n=1, called the integral root theorem.

Practice

Utilise the rational root theorem to listing all possible rational roots for the equation x³+2x-9=0.

Utilise the rational root theorem to factorize the following polynomial function:

f(x)=2x⁴-11xⁱⁱⁱ+4x²+14x-3

But factorize as much as you tin using the theorem.

First, use the theorem to find the possible rational roots.

Then plug those values into f(x) as 10 to see if they are roots.

Finally, employ the known roots to factorize equally much equally you can.

The focus of this lesson is using the theorem, just for those of yous that took information technology to the next level and used polynomial long or constructed partitioning, the quadratic is (x^two-5x+1).

How did you get on with the questions? Were there any parts you weren't confident with? Any questions you have tin be asked in the comments at the end of the lesson.

To Sum Up (Pun Intended!)

Rational numbers can be written equally a fraction of whole numbers, or as a decimal number that either has an cease, like 0.65, or a repeating pattern similar 0.16161616…

The roots of a polynomial f(x) are values of x that solve the equation f(ten)=0.

Equally the proper noun suggests, a rational root is the combination of a rational number with a root.

The rational root theorem, which is also chosen the rational zero theorem, says that any rational roots of the polynomial must be one of the following: