TRINOMIAL OF THE FORM X ^ 2 + BX + C (WITH EXAMPLES) - MATHS - 2025

Before learning to solve the trinomial of the form x ^ 2 + bx + c, and even before knowing the concept of a trinomial, it is important to know two essential notions; namely, the concepts of monomial and polynomial. A monomial is an expression of the type a * x ⁿ, where a is a rational number, n is a natural number and x is a variable.

A polynomial is a linear combination of monomials of the form a _n * x ⁿ + a _n-1 * x ^n-1 +… + a ₂ * x ² + a ₁ * x + a ₀, where each a _i, with i = 0,…, n, is a rational number, n is a natural number and a_n is nonzero. In this case the degree of the polynomial is said to be n.

A polynomial formed by the sum of only two terms (two monomials) of different degrees is known as a binomial.

Trinomials

A polynomial formed by the sum of only three terms (three monomials) of different degrees is known as a trinomial. The following are examples of trinomials:

• x ³ + x ² + 5x
• 2x ⁴ -x ³ +5
• x ² + 6x + 3

There are several types of trinomials. Of these, the perfect square trinomial stands out.

Perfect square trinomial

A perfect square trinomial is the result of squaring a binomial. For example:

• (3x-2) ² = 9x ² -12x + 4
• (2x ³ + y) ² = 4x ⁶ + 4x ³ y + y ²
• (4x ² -2y ⁴) ² = 16x ⁴ -16x ² y ⁴ + 4y ⁸
• 1 / 16x ² y ⁸ -1 / 2xy ⁴ z + z ² = (1 / 4xy ⁴) ² -2 (1 / 4xy ⁴) z + z ² = (1 / 4xy ⁴ -z) ²

Characteristics of grade 2 trinomials

Perfect square

In general, a trinomial of the form ax ² + bx + c is a perfect square if its discriminant is equal to zero; that is, if b ² -4ac = 0, since in this case it will have a single root and it can be expressed in the form a (xd) ² = (√a (xd)) ², where d is the already mentioned root.

A root of a polynomial is a number in which the polynomial becomes zero; in other words, a number that, when substituting for x in the polynomial expression, results in zero.

Resolving formula

A general formula to calculate the roots of a second-degree polynomial of the form ax ² + bx + c is the resolvent formula, which states that these roots are given by (–b ± √ (b ² -4ac)) / 2a, where b ² -4ac is known as the discriminant and is usually denoted by ∆. From this formula it follows that ax ² + bx + c has:

- Two different real roots if ∆> 0.

- A single real root if ∆ = 0.

- It has no real root if ∆ <0.

In what follows, only trinomials of the form x ² + bx + c will be considered, where clearly c must be a number other than zero (otherwise it would be a binomial). These types of trinomials have certain advantages when factoring and operating with them.

Geometric interpretation

Geometrically, the trinomial x ² + bx + c is a parabola which opens upward and has the vertex at point (-b / 2, -b ² /4 + c) of the Cartesian plane that x ² + bx + c = (x + b / 2) ² -b ² /4 + c.

This parabola cuts the Y axis at the point (0, c) and the X axis at the points (d ₁, 0) and (d ₂, 0); then d ₁ and d ₂ are the roots of the trinomial. It can happen that the trinomial has a single root d, in which case the only cut with the X axis would be (d, 0).

It could also happen that the trinomial does not have any real root, in which case it would not intersect the X axis at any point.

For example, x ² + 6x + 9 = (x + 3) ² -9 + 9 = (x + 3) ² is the parabola with vertex at (-3,0), which intersects the Y axis at (0, 9) and to the X axis at (-3,0).

Trinomial factoring

A very useful tool when working with polynomials is factoring, which consists of expressing a polynomial as a product of factors. In general, given a trinomial of the form x ² + bx + c, if it has two different roots d ₁ and d ₂, it can be factored as (xd ₁) (xd ₂).

If it has a single root d, it can be factored as (xd) (xd) = (xd) ², and if it has no real root, it remains the same; in this case it does not admit a factorization as a product of factors other than itself.

This means that, knowing the roots of a trinomial in the already established form, its factorization can be easily expressed, and as already mentioned above, these roots can always be determined using the resolvent.

However, there is a significant amount of this type of trinomials that can be factored without first knowing their roots, which simplifies the work.

The roots can be determined directly from the factorization without using the resolvent formula; these are the polynomials of the form x ² + (a + b) x + ab. In this case we have:

x ² + (a + b) x + ab = x ² + ax + bx + ab = x (x + a) + b (x + a) = (x + b) (x + a).

From this it is easily seen that the roots are –a and –b.

In other words, given a trinomial x ² + bx + c, if there are two numbers u and v such that c = uv and b = u + v, then x ² + bx + c = (x + u) (x + v).

That is, given a trinomial x ² + bx + c, it is first verified if there are two numbers such that multiplied they give the independent term (c) and added (or subtracted, depending on the case), they give the term that accompanies the x (b).

Not with all trinomials in this way this method can be applied; in which it is not possible, the resolution is used and the aforementioned applies.

Examples

Example 1

To factor the following trinomial x ² + 3x + 2, proceed as follows:

You must find two numbers such that when adding them the result is 3, and that when multiplying them the result is 2.

After making an inspection it can be concluded that the numbers searched are: 2 and 1. Therefore, x ² + 3x + 2 = (x + 2) (x + 1).

Example 2

To factor the trinomial x ² -5x + 6, we look for two numbers whose sum is -5 and their product is 6. The numbers that satisfy these two conditions are -3 and -2. Therefore, the factorization of the given trinomial is x ² -5x + 6 = (x-3) (x-2).

References

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