Factoring a cubic expression may seem like an intimidating task, but with the right approach, it can be broken down into manageable steps. The key to success lies in understanding the structure of a cubic expression and employing factorization techniques strategically. By following a systematic process, you can effectively factorize any cubic expression, revealing its underlying factors and simplifying the expression.
To begin the factorization process, it’s essential to identify any factors that can be easily extracted from the expression. This includes common factors among all terms and any factors that are perfect cubes. By factoring out these common elements, the expression becomes more manageable. The remaining expression, known as the depressed cubic, can then be further analyzed using various factorization methods. These methods include grouping, completing the cube, and using Vieta’s formulas to find rational roots. By applying these techniques effectively, you can successfully factorize the depressed cubic and obtain the complete factorization of the original cubic expression.
Throughout the factorization process, it’s crucial to verify your results by expanding the factored expression and comparing it to the original expression. This step ensures that the factorization is correct and that no errors have been made. By following a systematic approach, utilizing appropriate factorization techniques, and verifying your results, you can confidently navigate the factorization of cubic expressions, unlocking their underlying structure and simplifying them for further analysis.
Factoring Trinomials with a Lead Coefficient of 1
Factoring a cubic expression involves finding the factors of the expression that result in the original expression when multiplied together. Trinomials with a lead coefficient of 1 can be factored using specific techniques.
Method of Grouping
This involves grouping the terms of the trinomial into two groups. The first group should contain the first two terms, and the second group should contain the last term. Find the greatest common factor (GCF) of the terms in each group. Factor out the GCF from each group and then factor the remaining terms within each group.
For example, to factor the trinomial x³ – 5x² + 6x, group the terms: (x³ – 5x²) + 6x. The GCF of the first two terms is x², so factor it out: x²(x – 5) + 6x. Factor the remaining terms in each group: x²(x – 5) + 6x = x(x – 5)(x + 1).
Special Case: x³ – 1
This trinomial can be factored using the difference of cubes formula: x³ – 1 = (x – 1)(x² + x + 1).
Table of Common Factoring Cases for x³ + bx² + cx + d
| Coefficient of x³ | Coefficient of x² | Constant Term | Factorization |
|---|---|---|---|
| 1 | 0 | 1 | (x + 1)(x² – x + 1) |
| 1 | -1 | 1 | (x – 1)(x² + x + 1) |
| 1 | 1 | -1 | (x + 1)(x – 1)² |
Factoring Trinomials with a Lead Coefficient of -a
When dealing with trinomials with a lead coefficient of -a, the factoring process can be a bit different. Here’s a step-by-step guide to help you factor these trinomials:
Step 1: Find the factors of the constant term
Start by finding the factors of the constant term, which is the last number in the trinomial. These factors should be integers whose product equals the constant term and whose sum equals the coefficient of the middle term (the number in the middle of the trinomial).
Step 2: Split the middle term into two terms
Take the coefficient of the middle term and split it into two factors that multiply to the constant term. These factors should have a sum equal to the coefficient of the middle term.
Step 3: Rewrite the trinomial
Replace the middle term with the two terms you found in Step 2. Rewrite the trinomial so that it contains three terms.
Step 4: Factor by grouping
Group the first two terms together and the last two terms together. Factor out the greatest common factor (GCF) from each group.
Step 5: Check for common factors and simplify
Look for any common factors between the two groups. Factor out the common factor and simplify the expression. If there are no common factors, the trinomial is fully factored. Here’s an example of factoring a trinomial with a lead coefficient of -a:
| Original Trinomial | Factored Trinomial |
|---|---|
|
-x2 – 5x + 6 |
-(x – 3)(x + 2) |
In this example, the constant term is 6, whose factors are 1 and 6 or 2 and 3. The sum of 1 and 6 is 7, not equal to the coefficient of the middle term (-5). However, the sum of 2 and 3 is -5, which is equal to the coefficient of the middle term. So, we split -5 into -2 and -3, rewrite the trinomial as -x2 – 2x – 3x + 6, and factor by grouping to get the final factored form: -(x – 3)(x + 2).
Factoring Trinomials with a Common Factor
A trinomial is a polynomial with three terms. A common factor is a factor that is common to all the terms in a trinomial. To factor a trinomial with a common factor, first find the greatest common factor (GCF) of the three terms. Then factor out the GCF from each term.
For example, to factor the trinomial 12x^2 + 15x + 18, first find the GCF of the three terms. The GCF of 12x^2, 15x, and 18 is 3. Then factor out the GCF from each term:
“`
12x^2 + 15x + 18 = 3(4x^2 + 5x + 6)
“`
Now factor the trinomial inside the parentheses. The trinomial 4x^2 + 5x + 6 factors into (2x + 3)(2x + 2). Therefore, the fully factored form of 12x^2 + 15x + 18 is:
“`
12x^2 + 15x + 18 = 3(2x + 3)(2x + 2)
“`
Example 1
Factor the trinomial 6x^3 + 9x^2 – 12x.
Solution: The GCF of the three terms is 3x. Factor out the GCF from each term:
“`
6x^3 + 9x^2 – 12x = 3x(2x^2 + 3x – 4)
“`
Now factor the trinomial inside the parentheses. The trinomial 2x^2 + 3x – 4 factors into (2x – 1)(x + 4). Therefore, the fully factored form of 6x^3 + 9x^2 – 12x is:
“`
6x^3 + 9x^2 – 12x = 3x(2x – 1)(x + 4)
“`
Example 2
Factor the trinomial 4x^4 – 16x^2 + 12x.
Solution: The GCF of the three terms is 4x. Factor out the GCF from each term:
“`
4x^4 – 16x^2 + 12x = 4x(x^3 – 4x + 3)
“`
Now factor the trinomial inside the parentheses. The trinomial x^3 – 4x + 3 factors into (x – 1)(x^2 – 3x + 3). Therefore, the fully factored form of 4x^4 – 16x^2 + 12x is:
“`
4x^4 – 16x^2 + 12x = 4x(x – 1)(x^2 – 3x + 3)
“`
Example 3
Factor the trinomial 8x^5 – 12x^3 + 16x.
Solution: The GCF of the three terms is 4x. Factor out the GCF from each term:
“`
8x^5 – 12x^3 + 16x = 4x(2x^4 – 3x^2 + 4)
“`
Now factor the trinomial inside the parentheses. The trinomial 2x^4 – 3x^2 + 4 factors into (2x^2 – 1)(x^2 – 2). Therefore, the fully factored form of 8x^5 – 12x^3 + 16x is:
“`
8x^5 – 12x^3 + 16x = 4x(2x^2 – 1)(x^2 – 2)
“`
Factoring Trinomials by Grouping
Factoring trinomials by grouping involves identifying common factors in the first two and last two terms of the trinomial and then factoring out those common factors to create a GCF (greatest common factor). The resulting expression is then grouped into two binomials, and the common factor is factored out from each binomial to obtain the final factorization.
To factor a trinomial of the form ax2+bx+c using grouping, follow these steps:
- Identify the GCF of the first two terms and the last two terms.
- Factor out the GCF from each pair of terms.
- Group the terms into two binomials.
- Factor out the GCF from each binomial.
- Combine the like terms to obtain the final factorization.
Example 7: Factoring 2x3-8x2+6x
In this example, the GCF of the first two terms is 2x2, and the GCF of the last two terms is 2x. Factoring out the GCFs and grouping the terms yields:
| Step | Expression |
|---|---|
| 1 | 2x2(x – 4) + 2x(x – 4) |
| 2 | (2x2 + 2x)(x – 4) |
| 3 | 2x(x + 1)(x – 4) |
Therefore, the factorization of 2x3-8x2+6x is 2x(x + 1)(x – 4).
Factoring Trinomials Using the Difference of Cubes Formula
The difference of cubes formula is a factorization formula that can be used to factor trinomials of the form ax³ + bx² + cx + d. The formula is:
“`
a³ – b³ = (a – b)(a² + ab + b²)
“`
To factor a trinomial using the difference of cubes formula, follow these steps:
1. Set the trinomial equal to zero.
2. Factor the first two terms as a difference of cubes.
3. Factor the last two terms as a difference of squares.
4. Group the first two factors and the last two factors.
5. Factor out the greatest common factor from each group.
6. Multiply the factors from each group to get the final factorization.
For example, to factor the trinomial x³ – 8x² + 16x – 16, we would follow these steps:
Step 1: Set the trinomial equal to zero.
x³ – 8x² + 16x – 16 = 0
Step 2: Factor the first two terms as a difference of cubes.
(x³ – 8x²) = x²(x – 8)
Step 3: Factor the last two terms as a difference of squares.
(16x – 16) = 16(x – 1)
Step 4: Group the first two factors and the last two factors.
(x³ – 8x²) + (16x – 16) = 0
Step 5: Factor out the greatest common factor from each group.
x²(x – 8) + 16(x – 1) = 0
Step 6: Multiply the factors from each group to get the final factorization.
(x – 8)(x² + 16x + 16)(x – 16) = 0
Therefore, the factorization of x³ – 8x² + 16x – 16 is (x – 8)(x + 8)(x – 16).
Factoring Trinomials Using the Sum of Cubes Formula
The Sum of Cubes Formula states that a3 + b3 = (a + b)(a2 – ab + b2). This formula can be used to factor trinomials of the form x3 + y3.
Steps:
1. Identify the values of a and b in the trinomial.
2. Write the trinomial as (a + b)(a2 – ab + b2).
Example:
Factor the trinomial x3 + 8y3.
1. Identify that a = x and b = 2y.
2. Write the trinomial as (x + 2y)(x2 – 2xy + 4y2).
Special Case: Factoring Trinomials of the Form x3 – y3
The Sum of Cubes Formula can also be used to factor trinomials of the form x3 – y3. In this case, the formula is a3 – b3 = (a – b)(a2 + ab + b2).
Steps:
1. Identify the values of a and b in the trinomial.
2. Write the trinomial as (a – b)(a2 + ab + b2).
Example:
Factor the trinomial x3 – 27y3.
1. Identify that a = x and b = 3y.
2. Write the trinomial as (x – 3y)(x2 + 3xy + 9y2).
Factoring Trinomials with a Negative Constant Term
When factoring trinomials with a negative constant term, we can use the following steps:
10. Step-by-Step Guide to Factoring Trinomials with a Negative Constant Term
To factor trinomials with a negative constant term, follow these steps:
- Find two numbers that multiply to the constant term and add to the coefficient of the middle term. For example, if the constant term is -12 and the coefficient of the middle term is -3, we need to find two numbers that multiply to -12 and add to -3.
- Write these two numbers as binomials. In our example, the two numbers are -6 and 2, so we write them as -6x and 2x.
- Factor out the common factor from each binomial. In our example, the common factor is x, so we write it out as x(-6 + 2).
- Simplify the expression inside the parentheses. In our example, we get x(-4) = -4x.
- So, the factored trinomial is x(-6x + 2x) = x(-4x) = -4x2.
Here is a table summarizing the steps:
| Step | Action |
|---|---|
| 1 | Find two numbers that multiply to the constant term and add to the coefficient of the middle term. |
| 2 | Write these two numbers as binomials. |
| 3 | Factor out the common factor from each binomial. |
| 4 | Simplify the expression inside the parentheses. |
| 5 | Factor out the remaining common factor, if any. |
How to Factorise a Cubic Expression
A cubic expression is a polynomial of degree 3. It can be written in the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants. To factorise a cubic expression, you can use a combination of algebraic techniques, such as factoring by grouping, using the difference of squares formula, and using the sum or difference of cubes formula.
Here are the steps on how to factorise a cubic expression:
- Factor out any common factors from all the terms in the expression.
- If the expression has a negative coefficient for the x^3 term, factor out -1.
- Group the terms in the expression into two groups, (ax^3 + bx^2) and (cx + d).
- Factor each group separately. For the first group, use the difference of squares formula or the sum or difference of cubes formula.
- Multiply the two factors from step 4 together to get the factorised cubic expression.
People Also Ask
How do you factorise a cubic expression with a negative coefficient?
If the expression has a negative coefficient for the x^3 term, factor out -1. Then, proceed with the steps above.
How do you use the difference of squares formula to factorise a cubic expression?
The difference of squares formula is (a + b)(a – b) = a^2 – b^2. You can use this formula to factorise a cubic expression if the first two terms are a perfect square trinomial and the last two terms are a perfect square binomial.
How do you use the sum or difference of cubes formula to factorise a cubic expression?
The sum or difference of cubes formula is (a + b)(a^2 – ab + b^2) = a^3 + b^3 and (a – b)(a^2 + ab + b^2) = a^3 – b^3. You can use this formula to factorise a cubic expression if the first and last terms are perfect cubes.
