In this explainer, we will learn how to find the set of zeros of a quadratic, cubic, or higher-degree polynomial function.

Polynomial functions appear all throughout science and in many real-world applications. For example, a ball thrown in the air will follow a parabolic arc that can be modeled by a quadratic equation. In particular, the height of the ball from the ground will be a quadratic function. Therefore, if we want to determine how long it would take the ball to hit the ground, we will need to find the values where a quadratic function is equal to zero.

The input values of where a function outputs zero are called the zeros (or roots) of the function, and we can write this more formally as follows.

### Definition: Zeros or Roots of a Function

If , then we say that is a zero (or root) of the function .

For example, for the function , we can see that so is a root of this function.

There are a few methods of finding the roots of a function. For example, since we are looking for the values of where , we can sketch the graph , and then the points on the curve where the output is zero are the roots; these will be the -intercepts. To see this, consider the following graph of .

From the graph, we can see that , , and ; these are the roots of the function. Since there can be multiple roots of a function, we usually write them in a set called the set of roots of the function; in this case, the set of roots of is .

Not all functions have zeros. Consider the function . In this function, every output is always 1, so no input can ever give us 0. In fact, the constant function of the form will only have zeros if . When a function does not have any zeros, we can say the set of zeros is .

There are pros and cons for finding the zeros of a function graphically versus algebraically. In this explainer, we will focus on finding algebraic roots of polynomials since these will give us exact values for the roots.

We can find the zeros of polynomials by factoring. For example, consider the function . We can factor this by finding a pair of numbers that multiply to give 6 and add to give 5; we see that and . Hence,

Therefore, the zeros of the function are the solutions of the equation

On the left-hand side of this equation, we have a product that must be equal to zero. We note that for a product to be equal to zero, one of the factors must be equal to zero. Hence, either

We can solve both of these equations separately to get and as the zeros of the function. It is worth noting that we could have used the quadratic formula to find the roots of the equation, but we can only use this when we have a quadratic function.

We recall that we can factor some higher-order polynomials by grouping. For example, in the function , we can factor the first two terms to get and the last two terms to get . Since these share a factor, we can take this out of the expression to get

We can factor by using the difference between squares, which we recall states that we can factor a difference of squares as follows: . This gives

Therefore, the zeros of this function are the solutions to the equation

Solving each factor to be equal to zero, we get as the zeros of the function.

There are other methods of factoring a polynomial. One such method is to use a fact called the factor theorem together with polynomial division. The factor theorem tells us that if is a polynomial and , then must be a factor of the polynomial. It is worth noting that the reverse statement is also true: if is a factor of a polynomial , then .

We can use this to help us find the roots of a polynomial. If is a polynomial and we know that , then we know that is a factor of . In particular, this means we can use polynomial division to divide by , allowing us to factor .

Letβs see some examples of applying these techniques to determine the zeros of polynomials. We will start with a linear function.

### Example 1: Finding the Zeros of a Linear Function

Find the set of zeros of the function .

### Answer

We recall that is a zero of the function if . Therefore, to find the zeros for this function, we need to solve the equation

This is the equation

Multiplying through by 3 gives

We then add 4 to both sides of the equation

We see that the only zero of the function is 4. Hence, the set of zeros of the function is .

In our second example, we will determine the zeros of a quadratic function by factoring.

### Example 2: Finding the Zeros of a Monic Quadratic Function by Factoring

Find, by factoring, the zeros of the function .

### Answer

We recall that the zeros of a function are the input values such that . Therefore, to find the zeros of the given quadratic, we need to solve the equation

There are several different methods of doing this. For example, we can use the quadratic formula. However, we will fully factor the quadratic. Recall that to factor a quadratic , we need to find two numbers that multiply to give and add to give . In our case, we have and . We can list the possible factor pairs of :

35 | |

7 | |

5 | |

1 |

Of these, we see that . We can use these to rewrite the term in the quadratic as

Using this to rewrite the equation gives us

We can then factor the first two terms and last two terms separately:

We then take out the shared factor of to get

For the product of two numbers to be equal to zero, one of the factors must be equal to zero. Hence, either or . We can solve each equation separately.

First,

We subtract 7 from both sides of the equation to get

Second,

We add 5 to both sides of the equation to get

Therefore, the zeros of the quadratic are and 5.

In our next example, we will find the roots of a nonmonic quadratic by factoring.

### Example 3: Finding the Zeros of a Nonmonic Quadratic Function by Factoring

Find, by factoring, the zeros of the function .

### Answer

We recall that we say is a zero of when . Therefore, to find the zeros for this function, we need to solve the equation

We could do this by using the quadratic formula; however, we will use the grouping method. We recall that to factor a quadratic , we need to find a factor pair of that add to give . In this quadratic, we have , , and , so we need to find a pair of numbers that multiple to give and add to give 9. We notice that

To apply the grouping method to factor this quadratic, we use these two numbers to rewrite the second term in the quadratic as follows:

Hence, we can rewrite the quadratic equation as

We then factor the first two terms and second two terms separately:

We can take out the shared factor of :

For a product to be equal to zero, one of the factors must be equal to zero. Hence, either

We can solve each equation separately. First,

We add 5 to both sides of the equation to get then, we divide through by 3, which gives us

Second,

We subtract 8 from both sides of the equation to get then, we divide through by 3, which gives us

Therefore, the zeros of the function are and .

Thus far, we have only found the zeros of polynomials of degree 2 or less. In the remaining examples, we will find the zeros of polynomials of degree 3 or more.

### Example 4: Finding the Zeros of a Quartic by Factoring

Find the set of zeros of the function .

### Answer

We recall that the zeros or roots of a function are the input values that cause the function to output zero. Therefore, to find the zeros of we need to solve the equation ; this is the equation

We can solve this equation by factoring. We note that both terms share a factor of ; taking out this factor gives

If we then take out a factor of , we have

We can factor further by recalling the difference between squares result, which states that

Applying this with gives

We can then rewrite the equation:

Since , this is now the product of linear factors, so we cannot factor any further. We can now find the zeros of the equation by recalling that if a product is equal to zero, then one of the factors must be equal to zero. We can solve each of the factors being equal to zero separately. First,

We divide through by :

Then, we take the square root of both sides of the equation, noting that zero is the only root:

Second,

We add 5 to both sides of the equation:

Third,

We subtract 5 from both sides of the equation, which gives us the final zero of the function:

It is worth noting that since we were not asked to factor the polynomial fully, we could have found the values of that make each factor in the expression equal to zero directly. We would then have which we can solve by taking the square root of both sides of the equation, where we remember we will get a positive and a negative square root:

Similarly, we have

Writing these as a set gives us that the set of zeros of this function is .

### Example 5: Finding the Zeros of a Partially Factored Cubic Function

Find the set of zeros of the function .

### Answer

We recall that is a zero of the function if . Therefore, to find the zeros for this function, we need to solve the equation

This is the equation

Since both terms on the left-hand side of this equation share a factor of , we will do this by factoring. Taking out the shared factor, we have

We can factor this further by recalling that the difference between squares tells us that for any constant , . Setting , we have . Therefore, we can rewrite the equation as

Since this is three linear factors, we have factored the expression fully. For a product to be equal to zero, one of the factors must be equal to zero. Hence, we have

We can solve each equation separately. First,

We add 9 to both sides of the equation to get

Second,

We subtract 9 from both sides of the equation, which gives us

Third,

We add 2 to both sides of the equation to get

Therefore, we have three solutions: , , or 2.

It is worth noting that since we were not asked to factor the polynomial fully, we could have found the values of that make each factor in the expression equal to zero directly. We would then have which we can solve by taking the square root of both sides of the equation, where we remember we will get a positive and a negative square root:

Similarly, we have

These are the zeros of the function. Writing these in a set, we have that the set of zeros of is .

In our final example, we will use the factor theorem to determine the roots of a cubic polynomial.

### Example 6: Finding the Zeros of a Cubic Function

Given that and , find the other roots of .

### Answer

We start by recalling that the roots of a function are the values of such that . Since we are given that , we know that is a root of . One way of finding the roots of a polynomial is to factor the function fully.

Since we are given a root of the function, we can do this by recalling part of the factor theorem: if is a polynomial and , then is a factor of .

In our case, is a cubic polynomial, and we are told . So, setting , the factor theorem tells us that is a factor of . There are then a few different methods we can use to factor the cubic polynomial; we will show two of these.

First, we can factor by using the grouping method. Since we know that is a factor of , we can write . Distributing the parentheses gives us

Equating the coefficients of gives . We can then substitute this into the expression to get

Equating the coefficients of gives which we can solve for by subtracting 1 from both sides of the equation

Similarly, equating the constant terms gives

We can substitute these values into the equation to see

This confirms that we found the values of the coefficients correctly. We have

We can factor this further by noticing that and . This means that

Hence, we have shown

For , we have and for a product to be zero, one of the factors must be equal to zero; hence, , , or 3. Since the question asked for the other two roots, we have and .

Alternatively, we could have used polynomial division to factor from . We have

Thus,

We could then factor the quadratic in the same way and solve to find that the two other roots of are and .

Letβs finish by recapping some of the important points of this explainer.

### Key Points

- The zeros or roots of a polynomial are the values such that .
- If is a polynomial and , then is a factor of . The same statement is true in reverse: if is a factor of the polynomial , then .
- There are many techniques we can use to help us determine the roots of polynomials including the quadratic formula, factoring by grouping, and polynomial division.