In algebra, a trinomial is simply a polynomial that has three terms. Now, don't get confused—polynomials are just expressions that involve variables and coefficients, and can include operations like addition, subtraction, multiplication, or division. For example, take the expression $3x^2 + 7y^3 – 4xy$. Here, we have three terms: $3x^2$, $7y^3$, and $-4xy$. Each term is separated by a plus or minus sign, and they can include both variables and numbers. Understanding trinomials is important for exams like the SAT, ACT, or MCAT, where you might be asked to simplify or factor them. So, next time you see a trinomial, remember it's just a three-term algebraic expression with variables and operations in it!

What is a Trinomial?

A trinomial equation is a polynomial equation involving three terms. Note that 3 terms are always separated by addition or subtraction and not by multiplication or division. All quadratic equations are trinomials.

Example of Trinomial: \(1 + x + x^2, -1 + z^2 – z^2x^2, 3x + 2y – z\)

• Monomial: A polynomial with 1 term. Example of monomial: xy.
• Binomial: A polynomial with 2 terms. Example of binomial: x+y.
• Trinomial: A polynomial with 3 terms. Example of trinomial: \(x^2 + xz + 1\)

What is a Perfect Square Trinomial?

Perfect square trinomial is one of these polynomials that are “simple to factor.” An expression obtained from the square of a binomial equation is a perfect square trinomial. An expression is said to be a perfect square trinomial if it takes the form \(ax^2 + bx + c\) and satisfies the condition \(b^2 = 4ac\).

The perfect square formula takes the following forms:

• \((ax)^2 + 2abx + b^2 = (ax + b)^2\)
• \((ax)^2 −2abx + b^2 = (ax−b)^2\)

How to identify Perfect Square Trinomial?

The following steps can help you identify a perfect square trinomial:

• Check whether the first and last terms of the trinomial are perfect squares.
• Multiply the roots of the first and third terms together.
• Compare the middle terms with the result in step two
• If the first and last terms are perfect squares, and the middle term’s coefficient is twice the product of the square roots of the first and last terms, then the expression is a perfect square trinomial.

Example of perfect square trinomial: \(x^2+ 6x + 9, 25y^2 – 10y + 16\)

What is Quadratic Trinomial?

The solutions to the quadratic equation can be obtained by rewriting quadratic trinomial, which are polynomials with three terms, in a factored form with two binomial factors.

If the quadratic expression is of the form \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are numbers, then the quadratic trinomial will make sure that neither \(a, b\), nor \(c \)will be 0. All three of these letters will be a number other than 0.

Example of Quadratic Trinomial: \(2x^2 + 6x + 4\)

What are Trinomial Identities?

Following are the standard identities of trinomial:

1. Trinomial Identity 1: \((a + b)(a + c)(b + c) = (a + b + c)(ab + ac +bc) – abc\)
2. Trinomial Identity 2: \(a^2 + b^2 + c^2 = (a + b + c)^2 – 2 (ab + ac + bc)\)
3. Trinomial Identity 3: \(a^3 + b^3 + c^3 – 3abc = (a + b + c)(a^2 + b^2 + c^2 – ab – ac – bc)\)

What are Factoring Trinomial?

Learning how to factor a trinomial is an extremely important and useful algebra skill, but factoring trinomial can also be very tricky.

Consider the following trinomial \(ax^2 + bx + c\).

The first time is an \(x^2\) term, the second term is an \(x\) term, and the third term is a constant where \(a\) = the \(x^2\) term coefficient, \(b\) = the \(x\) term coefficient and \(c\) = the constant value.

How to Factor a Trinomial when \(a = 1\)?

Let’s learn how to factor a trinomial when \(a\), the leading coefficient is 1.

• Step 1: Identify the values for\(b\) and \(c\).
• Step 2: Find two numbers that ADD to \(b\) and MULTIPLY to \(c\).
• Step 3: Use the numbers you picked to write out the factors and check.

How to Factor a Trinomial when \(a \neq 1\)?

Let’s learn how to factor a trinomial when a, the leading coefficient is not equal to 1.

• Step 1: Identify the values for \(a, b\) and \(c\).
• Step 2: Find two numbers that ADD to \(b\) and MULTIPLY to \(ac\).
• Step 3: Use the numbers you picked to write out the factors and check.

Factoring Trinomial Formulas

Following standard identities used for trinomial factoring.

• Identity 1: \((a + b)^2 = a^2 + 2ab + b^2\)
• Identity 2: \((a – b)^2 = a^2 + b^2 – 2ab\)
• Identity 3: \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\)
• Identity 4: \((a – b)^3 = a^3 – b^3 – 3ab(a – b)\)

Factoring Trinomial using Identities

In order to factor a perfect square trinomial we use the following identities in reverse:

Identity 1: \((a + b)^2 = a^2 + 2ab + b^2\)

Identity 2: \((a – b)^2 = a^2 + b^2 – 2ab\)

Let us see how we do that with an example: Factor \(x^2+ 6x + 9\)

Step 1: We can rewrite the expression \(x^2 + 6x + 9\) in the form \(a^2 + 2ab + b^2\) as;

Step 2: \(x^2+ 6x + 9 ⟹ (x)^2 + 2(x)(3) + (3)^2\)

Step 3: Applying the formula of \((a + b)^2 = a^2 + 2ab + b^2\) to the expression gives;

Step 4: \(= (x + 3)^2\)

Step 5: \(= (x + 3)(x + 3)\)

Factoring Trinomial by Splitting Middle Term

Now let’s see how we can factor a non-perfect square or normal quadratic trinomial. We have seen the steps above. Let’s apply it to an example.

Factor \(x^2 + 9x + 20\)

where \(a = 1\),

Thus, \(x^2 + 9x + 20\)

Step 1: Identify the values for \(b\) and \(c\). Here \(b = 9\) and \(c = 20\)

Step 2: Find two numbers that ADD to \(b\) and MULTIPLY to \(c\). Here the numbers are 4 and 5.

Step 3: Use the numbers you picked to write out the factors and check. Thus, we can write the equation as \(x^2 + (5+4)x + (5\times4)\)

Step 4: \(= x^2 + 5x + 4x + (5\times4)\)

Step 5: Grouping the first two and last two terms, and find their GCF.

Step 6: \(= x^2 + 5x + 4x + (5\times4)\)

Step 7: \(= x(x+5) + 4(x+5)\)

Step 8: \(= (x+4)(x+5)\)

Factoring Trinomial using Greatest Common Factor

• GCF stands for the greatest common factor.
• The GCF for a polynomial is the largest monomial that divides each term of the polynomial.
• This is like using the distributive law in reverse.
• \(ab + ac = a({b + c})\).
• Simply examine each term to see if there is a characteristic common factor in all the terms.
• If so, we’ll factor the polynomial to get rid of it.

Example: \(8{x^4} – 4{x^3} + 10{x^2} = 2{x^2}({4{x^2} – 2x + 5} )\)

Solved Examples on Trinomial

Let’s get some practice on the concepts through some examples.

Example 1: Factor the trinomial \(9x^2 + 4x -5\) and \(2x^2 + 9x – 5\)

Solution: \(9x^2 + 4x -5\)

Writing splitting the middle term.

=\(9x^2 +9x – 5x -5\)

Grouping the first two and last two terms to find the common factor.

=\(9x(x + 1) – 5(x +1)\)

Factoring using the greatest common factor.

=\((9x-5)(x+1)\)

\(2x^2 + 9x – 5\)

Writing splitting the middle term.

=\(2x^2 + 10x – x – 5\)

Grouping the first two and last two terms to find the common factor.

=\(2x(x + 5) – 1(x + 5)\)

Factoring using the greatest common factor.

=\((2x-1)(x+5)\)

Example 2: Find the solution of the trinomial: \(x^4 – 2x^2 = 15\)

Solution: Write the polynomial in standard form.

\(x^4 – 2x^2 = 15\)

Equate it to zero

\(x^4 – 2x^2 – 15 = 0\)

It is difficult to solve for a fourth power root hence, we substitute \(x^2 = a\). This will make the polynomial quadratic easier to solve.

\(a^2 – 2a – 15 = 0\)

Writing the middle terms such that it becomes easier to factorize the polynomial. See Factoring Quadratic Equations for details.

\(a^2 – 5a + 3a – 15 = 0\)

Grouping the polynomial into two groups of binomials: The first two terms and the last two terms.

\((a^2 – 5a) + (3a – 15) = 0\)

Find the greatest common factor of each group and write them as their product i.e. reverse of the distributive law.

\(a(a – 5) + 3(a – 5) = 0\)

The two groups in turn have a common factor, we continue factorizing.

\((a+3)(a – 5) = 0\)

This is the simplest factor form or prime factorization of the polynomial. Now we substitute \(x^2 = a\).

Thus, \((x^2+3)(x^2 – 5) = 0\)

\((x^2+3)(x – 5)(x + 5) = 0\) …….. (as \(a^2 – b^2 = (a + b)(a – b)\))

For this equation to exist, either of these factors must be equal to zero.

Hence, \((x^2+3) = 0\) OR \((x – 5) = 0\) OR \((x + 5) = 0\)

\(x^2=-3\), thus \(x=\pm\sqrt{3}i\)

\((x – 5) = 0\), thus \(x = 5 \)

\((x + 5) = 0\), thus \(x = -5\)

Thus, the polynomial will have four solutions at \(x= -\sqrt{3}i, +\sqrt{3}i, -5 and +5\). The number of zeros is equal to the degree of the polynomial.

Example 3: Factorise the trinomial \(x^2+12x+36\).

Solution: \(x^2+12x+36\)

This is a perfect square trinomial. We use the trinomial formula to factorize the perfect square polynomial.

\((a + b)^2 = a^2 + 2ab + b^2\) we reverse this formula.

\(x^2+12x+36\) can be written as \(x^2+2(6)x+6^2\)

Thus, the factors are: \((x+6)(x+6)\)

Conclusion

In conclusion, understanding trinomials is essential for mastering algebra, especially when preparing for exams like the SAT, ACT, and MCAT. Whether it's recognizing a trinomial, factoring it, or identifying perfect square trinomials, these skills will help you tackle a wide range of algebraic problems. With practice, you'll get the hang of factoring, applying identities, and solving quadratic equations, all of which are crucial for both test prep and real-world math applications!