Problem 88 - Elementary Algebra

Question

For the following problems, factor, if possible, the trinomials. $$ x^{2}+x+1 $$

Step-by-Step Solution

Answer

Answer: No, the trinomial $$x^2 + x + 1$$ cannot be factored, as its discriminant is less than zero. The simplest form of this expression is $$x^2 + x + 1$$.

Step 1: Identify the coefficients of the trinomial

First, identify the coefficients of the trinomial. The given trinomial is $$x^2 + x + 1$$, with coefficients a=1, b=1, and c=1.

Step 2: Check if the trinomial is factorable

To find out if the trinomial is factorable, we can use the discriminant. The discriminant is given by the formula $$D = b^2 - 4ac$$. If D>0, the trinomial is factorable. If D≤0, the trinomial is not factorable. Plugging in the values of a, b, and c into the discriminant formula, we get: $$D = (1)^2 - 4 \cdot (1) \cdot (1) = 1 - 4 = -3$$ Since D=-3<0, the trinomial is not factorable. Since the given trinomial is not factorable, the original expression $$x^2 + x + 1$$ is the simplest form.

Key Concepts

Understanding the Discriminant

The discriminant is a powerful tool that helps determine whether a polynomial can be factored into real linear factors. It comes from the quadratic formula used to solve quadratic equations: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The part under the square root, \(b^2 - 4ac\), is the discriminant.

• If the discriminant is greater than zero, the trinomial has two distinct real roots and can usually be factored.
• If it equals zero, the trinomial has one repeated real root, also allowing it to be factorable.
• If it's less than zero, the roots are not real numbers, meaning the polynomial cannot be factored over the real numbers.

In the example \(D = 1 - 4 = -3\), the negative discriminant indicates no real factorization is possible.

Decoding Coefficients

Coefficients are the numerical factors of the terms in a polynomial expression. In any trinomial like \(ax^2 + bx + c\),

• \(a\) is the coefficient of the squared term,
• \(b\) is the coefficient of the linear term, and
• \(c\) is the constant term.

Knowing these coefficients allows you to analyze and solve polynomial equations, as they are integral parts of the discriminant and other solving methods.
For the trinomial \(x^2 + x + 1\), the coefficients are \(a = 1\), \(b = 1\), and \(c = 1\). Identifying these correctly is the first step to efficiently applying the quadratic formula or determining factorability using the discriminant.

Insight into Polynomial Expressions

A polynomial expression is a mathematical expression that involves a sum of powers of one or more variables multiplied by coefficients. They are often expressed as

• Monomials, which have a single term, like \(3x^2\).
• Binomials, which have two terms, like \(x + 1\).
• Trinomials, which have three terms, like \(x^2 + x + 1\).

When factoring polynomial expressions, particularly trinomials, we look for two expressions to multiply together to reproduce the original polynomial.
However, not all trinomials can be factored simply. In cases like \(x^2 + x + 1\), with a negative discriminant, we see that factoring over the real numbers isn't possible.
Recognizing such scenarios is important, as it may direct us to seek other methods or confirm that the original form is already simplified.