** In math, trinomial is a type of polynomial.**** Trinomial is an expression containing 3 terms. Quadratic function is a widely used trinomial. Here, we concentrate on the trinomial- quadratic function, since it has wide range of application. Specified expression consists of three expressions to be into the regular form of ax ^{2} + bx + c.**

**Polynomial with three terms is called as trinomial.**

** **

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A polynomial consisting of 3 terms is a trinomial, such as ax^{2} + bx + c where ax^{2}, bx, and c are 3 terms.

**For Example:** 3x + 4y^{2} - 1, 5m - 7n + x^{2} are the trinomials.

A polynomial is prime, if the only factors of a polynomial are 1 and itself. A prime trinomial can not be written as the product of lower degree polynomials or we can say that prime trinomials can not be factored.

** Trinomial is an expression containing 3 unlike terms. Factoring a trinomial means to originate math operation of multiplication in a reverse method.** Factoring is a reverse process of the FOIL method. **Let us establish some instance problems for factoring trinomials.**

### Factoring Quadratic Trinomials

Quadratic trinomial is a trinomial having highest degree of 2. Quadratic trinomial can be factored using methods like, factoring by grouping, completing square or by using quadratic formula.

### Factoring Trinomials Formula

Factoring trinomials formula for quadratic trinomial, ax^{2} + bx + c, is given below:

x = $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

### Factoring Trinomials Examples

Let us factorize 4x^{2} + 11x - 3

Given 4x^{2} + 11x - 3

4x^{2} + 11x - 3 = 4x^{2} + 12x - x - 3 (factoring by grouping method)

= 4x(x + 3) - (x + 3)

= (4x - 1)(x + 3)

Therefore, factors of 4x^{2} + 11x - 3 are (4x - 1) and (x + 3).

Perfect square trinomial is the product of two identical binomials. A trinomial of the form a^{2} + 2ab + b^{2 }or a^{2} - 2ab + b^{2} is a perfect square trinomial with factor (a + b)^{2} or (a - b)^{2} respectively.

Let us find the factors of x^{2} + 10x + 25

→ Read More Multiplying of trinomials can be done by distributive property. A trinomial can be distributed over another expression. Multiply each term from the first trinomial times each term of the second trinomial. Then the entire expression is simplified, combining anything that can be combined.

Let us multiply two trinomials, 2x + y + 2 and 4y + x - 5

2x + y + 2 ---------------- First Trinomial

4y + x - 5 ---------------- Second Trinomial

**Step 1:**

= 8xy + 2x^{2 } - 10x .........................(i)

Multiply the second term of the first trinomial with each term of the second trinomial.

y * (4y + x - 5) = y * 4y + y * x - y * 5

= 4y^{2} + yx - 5y ..........................(ii)

Multiply the third term of the first trinomial with each term of the second trinomial.

2 * (4y + x - 5) = 2 * 4y + 2 * x + 2 * -5

= 8y + 2x - 10 ......................(iii)

^{2 } + 4y^{2 } - 8x - 5y - 10

=> Multiplication of 2x + y + 2 and 4y + x - 5 is 9xy + 2x^{2 } + 4y^{2 } - 8x - 5y - 10.

→ Read More Cubic trinomial is a trinomial having 3 terms and of degree 3. For example, x^{3} + 2x^{2} + 12 is a cubic trinomial, where x^{3}, 2x^{2}, 12 are 3 terms and degree = 3.

### Factoring Cubic Trinomials

Factor the cubic trinomials of the form ax^{3} + bx^{2} + cx

Let us factor the trinomial, 2x^{3 }+ 6x^{2} - 36x.

**Step 1:**

Factor the greatest common factor of the trinomial.

2x^{3 }+ 6x^{2} - 36x = 2x(x^{2 }+ 3x - 18)

Step 2:

Factor the quadratic trinomial, x^{2 }+ 3x - 18

=> x^{2 }+ 3x - 18 = x^{2 }+ 6x - 3x - 18

= x(x + 6) - 3(x + 6)

= (x - 3)(x + 6)

Step 3:

The factored form of the cubic trinomial by multiplying the GCF by the factored form of the polynomial.

Factors of 2x^{3 }+ 6x^{2} - 36x = 2x (x - 3)(x + 6).

### Solved Examples

**Question 1: **Factor the trinomial x^{2} - x - 30.

** Solution: **

**Question 2: **Factor the trinomial x^{2 }- 2x - 8.

** Solution: **

A polynomial is prime, if the only factors of a polynomial are 1 and itself. A prime trinomial can not be written as the product of lower degree polynomials or we can say that prime trinomials can not be factored.

Quadratic trinomial is a trinomial having highest degree of 2. Quadratic trinomial can be factored using methods like, factoring by grouping, completing square or by using quadratic formula.

Factoring trinomials formula for quadratic trinomial, ax

x = $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Let us factorize 4x

Given 4x

4x

= 4x(x + 3) - (x + 3)

= (4x - 1)(x + 3)

Therefore, factors of 4x

Perfect square trinomial is the product of two identical binomials. A trinomial of the form a

Let us find the factors of x

x^{2} + 10x + 25 |
The first term, x^{2} , and the last term, 25 = 5^{2} are perfect squares. |

x^{2} + 10x + 5^{2} |
The square terms are rewritten and compared with a^{2} + 2ab + b^{2} |

2ab = 2 * x * 5 = 10x |
The middle term of the trinomial is verified. |

x^{2} + 10x + 5^{2} = (x + 5)^{2} |
The formula, a^{2} + 2ab + b^{2} = (a + b)^{2} is applied to write the factors |

→ Read More Multiplying of trinomials can be done by distributive property. A trinomial can be distributed over another expression. Multiply each term from the first trinomial times each term of the second trinomial. Then the entire expression is simplified, combining anything that can be combined.

Let us multiply two trinomials, 2x + y + 2 and 4y + x - 5

2x + y + 2 ---------------- First Trinomial

4y + x - 5 ---------------- Second Trinomial

Multiply the first term of the first trinomial with each term of second trinomial.

2x * (4y + x - 5) = 2x * 4y + 2x * x - 2x * 5= 8xy + 2x

Multiply the second term of the first trinomial with each term of the second trinomial.

y * (4y + x - 5) = y * 4y + y * x - y * 5

= 4y

Multiply the third term of the first trinomial with each term of the second trinomial.

2 * (4y + x - 5) = 2 * 4y + 2 * x + 2 * -5

= 8y + 2x - 10 ......................(iii)

**Step 2:**

Add (1), (2) and (3)

8xy + 2x^{2 } - 10x + 4y^{2} + yx - 5y + 8y + 2x - 10

Group the like terms and combine them.

= 8xy + xy + 2x^{2 } + 4y^{2 } - 10x + 2x - 5y - 10

=> Multiplication of 2x + y + 2 and 4y + x - 5 is 9xy + 2x

→ Read More Cubic trinomial is a trinomial having 3 terms and of degree 3. For example, x

Factor the cubic trinomials of the form ax

ax^{3} + bx^{2} + cx = x(ax ^{2} + bx + c) |
Factor the greatest common factor of the trinomial. |

ax^{2} + bx + c = (x + A)(x + B) |
Factor the quadratic trinomial |

Factors of ax^{3} + bx^{2} + cx = x(x + A)(x + B) |
Multiply the GCF by the factored form of the polynomial. |

Let us factor the trinomial, 2x

Factor the greatest common factor of the trinomial.

2x

Step 2:

Factor the quadratic trinomial, x

=> x

= x(x + 6) - 3(x + 6)

= (x - 3)(x + 6)

Step 3:

The factored form of the cubic trinomial by multiplying the GCF by the factored form of the polynomial.

Factors of 2x

Trinomial can be written as the product of there factors. And, by finding the factors we can find the unknown values of solve the expression much easily. How to solve trinomial can be discussed using the below examples.

**Step 1:** the given trinomial is x^{2 }- x - 30

**Step 2:** Factorize the given trinomial

x^{2 }- 6x + 5x - 30

**Step 3:** Factors are,

(x - 6)(x + 5)

So, the solution is x^{2} - x - 30 = (x - 6)(x + 5)

**Step 1:** Given trinomial is x^{2} - 2x - 8

**Step 2:** Factorize the given trinomial

x^{2} + 2x - 4x - 8

**Step 3: **Factors are,

(x + 2)(x - 4)

So, the solution is x^{2 }- 2x - 8 = (x + 2)(x - 4)

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