Factoring a cubic function is a common task in algebra. However, it can be a daunting one, especially if you don’t know where to start. In this article, we will provide you with a step-by-step guide on how to factorise a cubic function. We will also provide some tips and tricks to make the process easier. However, before we start, let’s quickly review what a cubic function is.
A cubic function is a polynomial function of degree 3. It has the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants. Cubic functions can be factorised into a product of three linear factors. For example, the cubic function f(x) = x³ – 3x² + 2x – 6 can be factorised as f(x) = (x – 1)(x – 2)(x + 3).
Now that we have a basic understanding of cubic functions, let’s take a closer look at how to factorise them. There are several different methods that you can use to factorise a cubic function. In this article, we will focus on the most common method, which is known as the sum of cubes factorisation method. This method is based on the fact that any cubic function can be written as the sum of two cubes. For example, the cubic function f(x) = x³ – 3x² + 2x – 6 can be written as f(x) = (x³) – (2x³ + 3x²) + (2x²) – (6x) + 6 = (x³ – 2x³) + (3x² – 2x²) + (2x – 6x) + 6 = (x³ – 2x³) + (x² – x²) + (x – 2x) + 6 = (x – 2)(x² + x – 3).
Understanding Cubic Functions
Cubic functions are polynomial functions of degree 3, which means that they are expressions that consist of a constant term, a linear term, a quadratic term, and a cubic term. The general form of a cubic function is ax³ + bx² + cx + d, where a, b, c, and d are real numbers and a is not equal to 0.
Cubic functions are often used to model real-world phenomena, such as the motion of objects under the influence of gravity, the growth of populations, and the cooling of objects. They can also be used to solve a variety of problems, such as finding the roots of a polynomial equation or finding the maximum or minimum value of a function.
The graph of a cubic function is a parabola that opens up or down. The shape of the parabola depends on the values of the coefficients a, b, c, and d. For example, if a is positive, the parabola will open up, and if a is negative, the parabola will open down.
Properties of Cubic Functions
Cubic functions have a number of properties that are unique to them. These properties include:
- The graph of a cubic function is a parabola that opens up or down.
- The x-intercepts of a cubic function are the roots of the corresponding polynomial equation.
- The y-intercept of a cubic function is the value of d.
- The maximum or minimum value of a cubic function occurs at the vertex of the parabola.
Identifying the Coefficients
The first step in factoring a cubic function is to identify its coefficients. The coefficients are the numerical constants that accompany the variables in the function. In the general form of a cubic function, ax³ + bx² + cx + d, the coefficients are a, b, c, and d.
It is important to note that the coefficient of the x³ term is always 1. This is because a cubic function is defined by having a variable raised to the third power.
To identify the coefficients of a cubic function, simply compare it to the general form. For example, if you are given the function x³ – 2x² + 5x – 3, the coefficients would be:
a = 1
b = -2
c = 5
d = -3
Once you have identified the coefficients of the cubic function, you can begin the process of factoring it.
Isolating a Factor
Let’s take a cubic function in its factored form,
f(x) = (x - a)(x^2 + bx + c).
Notice that the linear factor
(x - a)
is placed first, followed by a quadratic factor
(x^2 + bx + c).Our goal is to isolate the linear factor
(x - a)on one side of the equation. To do this, we'll multiply both sides by the denominator of the linear factor, which is
(x - a):
f(x)(x - a) = (x - a)(x^2 + bx + c)This gives us a new equation:
f(x)(x - a) = x^3 + bx^2 + cx - ax^2 - abx - acSimplifying the right-hand side, we get:
f(x)(x - a) = x^3 + (b-a)x^2 + (c-ab)x - acNow, we have successfully isolated the linear factor
(x - a)on the left-hand side of the equation. This allows us to proceed with factoring the remaining quadratic factor
(x^2 + (b-a)x + (c-ab))using the quadratic formula or other appropriate techniques.
Using the Rational Root Theorem
The Rational Root Theorem is a powerful tool for finding rational roots of a polynomial. It states that if a polynomial with integer coefficients has a rational root p/q in simplest form, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Finding Possible Rational Roots
The first step to factorising a cubic function using the Rational Root Theorem is to find the possible rational roots. To do this, we need to list the factors of the constant term and the leading coefficient.
For example, consider the cubic function f(x) = x^3 - 2x^2 - 5x + 6. The constant term is 6, which has factors 1, 2, 3, and 6. The leading coefficient is 1, which has factors 1 and -1. Therefore, the possible rational roots are:
| Factors of the constant term | Factors of the leading coefficient | Possible rational roots |
|---|---|---|
| 1 | 1 | ±1 |
| 2 | 1 | ±2 |
| 3 | 1 | ±3 |
| 6 | 1 | ±6 |
| 1 | -1 | ±1/1 |
| 2 | -1 | ±2/1 |
| 3 | -1 | ±3/1 |
| 6 | -1 | ±6/1 |
Testing the Roots
The next step is to test the possible rational roots. We can do this by plugging each root into the cubic function and checking if the result is zero.
For example, to test the root x = 1, we plug it into the cubic function:
```
f(1) = 1^3 - 2(1)^2 - 5(1) + 6
= 1 - 2 - 5 + 6
= 0
```
Since f(1) = 0, we know that x = 1 is a root of the cubic function.
We can continue testing the other possible rational roots until we find one that works. In this case, we find that x = 2 is also a root.
Factoring the Cubic Function
Once we have found the rational roots, we can factor the cubic function using the following formula:
```
f(x) = (x - r1)(x - r2)(x - r3)
```
where r1, r2, and r3 are the three rational roots.
For the cubic function f(x) = x^3 - 2x^2 - 5x + 6, the rational roots are x = 1 and x = 2. Therefore, we can factor the cubic function as follows:
```
f(x) = (x - 1)(x - 2)(x + 3)
```
Factoring by Grouping
Factoring by grouping involves breaking down a cubic function into smaller groups that can be factored and simplified individually. To factor a cubic function using this method, follow these steps:
1. Identify Groups
Divide the function into three groups, each containing one term with x, one term with x^2, and one constant term.
2. Factor Each Group
Factor each group as a quadratic expression. If the group has no real factors, leave it as is.
3. Combine Factors
Multiply the factors from each group to obtain the complete factorization of the cubic function.
4. Special Case: Common Factors
If there is a common factor among all the groups, factor it out first and then proceed with the above steps.
5. Example
Consider the cubic function: x^3 - 5x^2 + 6x - 18
| Group 1 | Group 2 | Group 3 |
|---|---|---|
| x^3 | -5x^2 | +6x |
| -5x^2 | -18 |
Group 1 is a single term and cannot be factored further.
Group 2 can be factored as -5x(x - 1).
Group 3 can be factored as 2(3x-9)=2*3(x-3).
Combining these factors, we get:
```
x^3 - 5x^2 + 6x - 18 = x^3 - 5x(x - 1) + 2*3(x - 3)
```
Factoring Perfect Cubes
A perfect cube is a number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it can be expressed as 23. The process of factoring a perfect cube involves expressing it as the product of three identical factors. This can be achieved using the following steps:
1. Find the cube root of the number. This is the number that, when multiplied by itself three times, gives the original number.
2. Raise the cube root to the power of 3. This gives you the original number.
3. Factor the cube root three times. This gives you the three identical factors of the perfect cube.
For example, to factor the perfect cube 8, we follow these steps:
```html
| 1. Cube root of 8 = 2 |
| 2. 23 = 8 |
| 3. (2)(2)(2) = 8 |
```
Therefore, the factors of 8 are 2, 2, and 2.
Here are some additional examples of factoring perfect cubes:
- 27 = 33 = (3)(3)(3)
- 64 = 43 = (4)(4)(4)
- 125 = 53 = (5)(5)(5)
Factoring Trinomials
A trinomial is a polynomial with three terms. To factor a trinomial, we need to find two binomials that multiply to give the original trinomial. For example, the trinomial x^2 + 5x + 6 can be factored as (x + 2)(x + 3). The constant terms 2 and 3 add up to 5 and multiply to give 6, the constant term in the original trinomial.
There is a shortcut method for factoring trinomials when the coefficient of the x2-term is 1, as in x^2 + bx + c. We can use the following steps:
- Find two numbers whose product is c and whose sum is b.
- Rewrite the middle term bx as the sum of these two numbers.
- Factor out the common factor from the two terms.
For example, to factor x^2 + 5x + 6, we find that 2 and 3 have a product of 6 and a sum of 5. We can then rewrite the trinomial as x^2 + 2x + 3x + 6 and factor out the common factor x to get (x + 2)(x + 3).
7. Special Cases
There are a few special cases of trinomials that can be factored easily. These include:
- The difference of squares, which can be factored as (a + b)(a - b).
- The perfect square trinomial, which can be factored as (a + b)2.
- The cube of a binomial, which can be factored as (a + b)3.
For example, the trinomial x2 - 4 can be factored as (x + 2)(x - 2) because it is the difference of squares. The trinomial x2 + 6x + 9 can be factored as (x + 3)2 because it is a perfect square trinomial. The trinomial x3 + 3x2 + 3x + 1 can be factored as (x + 1)3 because it is a cube of a binomial.
| Special Case | Factored Form |
|---|---|
| Difference of squares | (a + b)(a - b) |
| Perfect square trinomial | (a + b)2 |
| Cube of a binomial | (a + b)3 |
Factoring a Sum of Cubes
A sum of cubes can be factored using the following formula:
```
a³ + b³ = (a + b)(a² - ab + b²)
```
For example, to factor the sum of cubes x³ + 8, we can use the following steps:
1. Find the cube root of each term: ∛x³ = x and ∛8 = 2.
2. Write the two terms as (x + 2) and (x² - 2x + 4).
3. Multiply the two terms together to get x³ + 8.
Therefore, the factorization of x³ + 8 is (x + 2)(x² - 2x + 4).
We can also use this formula to factor a difference of cubes:
```
a³ - b³ = (a - b)(a² + ab + b²)
```
For example, to factor the difference of cubes x³ - 8, we can use the following steps:
1. Find the cube root of each term: ∛x³ = x and ∛8 = 2.
2. Write the two terms as (x - 2) and (x² + 2x + 4).
3. Multiply the two terms together to get x³ - 8.
Therefore, the factorization of x³ - 8 is (x - 2)(x² + 2x + 4).
Special Cases
There are some special cases that can be factored more easily:
| Case | Factorization |
|---|---|
| a³ + b³ | (a + b)(a² - ab + b²) |
| a³ - b³ | (a - b)(a² + ab + b²) |
| a³ + 2a²b + ab² | a(a + b)² |
| a³ - 2a²b + ab² | a(a - b)² |
Factoring a Difference of Cubes
A difference of cubes is a polynomial of the form a³ - b³, where a and b are real numbers. To factor a difference of cubes, we use the following formula:
a³ - b³ = (a - b)(a² + ab + b²)
For example, to factor the difference of cubes 8x³ - 125, we would use the following steps:
- Find the cube roots of a and b. In this case, the cube root of 8x³ is 2x and the cube root of 125 is 5.
- Write the difference of cubes as (a - b)(a² + ab + b²). In this case, we would write 8x³ - 125 as (2x - 5)(4x² + 10x² + 25).
Therefore, the factored form of 8x³ - 125 is (2x - 5)(4x² + 10x² + 25).
Special Case: When a = 9
When a = 9, the difference of cubes formula becomes:
9 - b³ = (3 - b)(9 + 3b + b²)
This formula can be used to factor any difference of cubes that has a leading coefficient of 9. For example, to factor the difference of cubes 9 - 27x³, we would use the following steps:
- Rewrite the difference of cubes as 9 - (27x)³.
- Apply the difference of cubes formula with a = 3 and b = 27x.
- Simplify the result.
Therefore, the factored form of 9 - 27x³ is (3 - 27x)(3 + 9x + 81x²).
Here is a table summarizing the factoring of a difference of cubes with general coefficients and the special case when a = 9:
| General Coefficients | a = 9 |
|---|---|
| a³ - b³ | 9 - b³ |
| (a - b)(a² + ab + b²) | (3 - b)(9 + 3b + b²) |
Verifying the Factorisation
Once you have factorised a cubic function, you can verify your answer by expanding the brackets and simplifying the expression. The result should be the original cubic function.
For example, if you have factorised the cubic function $f(x) = x^3 - 2x^2 - 5x + 6$ as $f(x) = (x - 2)(x^2 + 2x - 3)$, you can verify your answer as follows:
| $$\begin{split}f(x) &= (x - 2)(x^2 + 2x - 3)\\ &= x^3 + 2x^2 - 3x - 2x^2 - 4x + 6\\ &= x^3 - 2x^2 - 5x + 6\end{split}$$ |
You can see that the expanded expression is the same as the original cubic function, which means that the factorisation is correct.
Here are some tips for verifying the factorisation of a cubic function:
- Use the FOIL method to multiply out the brackets.
- Simplify the expression carefully, combining like terms.
- Check that the result is the same as the original cubic function.
How To Factorise A Cubic Function
To factorise a cubic function, you can use the following steps:
- Find the roots of the function.
- Factor out the roots using the factor theorem.
- Find the remaining factor by dividing the original function by the factored expression.
For example, to factorise the cubic function f(x) = x^3 - 8x^2 + 19x - 12, you would first find the roots of the function. The roots of the function are x = 1, x = 2, and x = 6.
You would then factor out the roots using the factor theorem. The factor theorem states that if a polynomial f(x) has a root at x = a, then f(x) can be factored as f(x) = (x - a) * g(x), where g(x) is a polynomial of degree one less than the degree of f(x).
Using the factor theorem, you can factor out the roots of f(x) as follows:
```
f(x) = (x - 1) * (x - 2) * (x - 6)
```
You would then find the remaining factor by dividing the original function by the factored expression. Dividing f(x) by (x - 1) * (x - 2) * (x - 6), you get:
```
f(x) = (x - 1) * (x - 2) * (x - 6) * (x - 3)
```
Therefore, the factorised form of f(x) is:
```
f(x) = (x - 1) * (x - 2) * (x - 6) * (x - 3)
```
People Also Ask
How do you find the roots of a cubic function?
To find the roots of a cubic function, you can use the following methods:
- The rational root theorem
- The cubic formula
- Numerical methods, such as the bisection method or the Newton-Raphson method
What is the factor theorem?
The factor theorem states that if a polynomial f(x) has a root at x = a, then f(x) can be factored as f(x) = (x - a) * g(x), where g(x) is a polynomial of degree one less than the degree of f(x).
How do you divide polynomials?
To divide polynomials, you can use the following methods:
- Long division
- Synthetic division
