91Ó°ÊÓ

To start with factoring any polynomial, the first and foremost step is to identify the Greatest Common Factor (GCF). The GCF is the largest factor that divides all the terms of the polynomial. In the trinomial given in the exercise, which is \(4x^4 + 2x^3 + x^2\), look closely at the coefficients and the variables.

• The coefficients are 4, 2, and 1.
• The variable parts are \(x^4\), \(x^3\), and \(x^2\).

The common factor in all the terms is \(x^2\), as each term contains at least two x's. Hence, the GCF is \(x^2\). This step simplifies the polynomial, making it easier to handle.

Once the GCF is identified, the next step in polynomial factoring is to factor out the GCF from the polynomial. This simplifies the polynomial into a more manageable form. Given the polynomial \(4x^4 + 2x^3 + x^2\), we factor out the GCF, which we identified as \(x^2\): \[ 4x^4 + 2x^3 + x^2 = x^2(4x^2 + 2x + 1) \] By doing this, we've simplified the polynomial to \(x^2\) times the quadratic expression \(4x^2 + 2x + 1\). The simplified form is much easier to work with, especially when checking for further factoring.

Now, let's focus on quadratic expressions. A quadratic expression is generally of the form \(ax^2 + bx + c\). In our factored expression, the quadratic part is \(4x^2 + 2x + 1\). The next step is to check if this quadratic expression can be factored further. Some quadratics can be factored into two binomials. However, in our case, \(4x^2 + 2x + 1\) does not factor further over the real numbers. To verify this, we can use the discriminant method: \[ \text{Discriminant} = b^2 - 4ac \] Here,

• \(a = 4\),
• \(b = 2\),
• \(c = 1\).

Substituting these values: \[ \text{Discriminant} = 2^2 - 4(4)(1) = 4 - 16 = -12 \] Since the discriminant is negative, there are no real roots, meaning it doesn't factor further over the set of real numbers. Thus, the factored form of the polynomial remains \(x^2(4x^2 + 2x + 1)\). This comprehensive understanding of quadratic expressions highlights why certain quadratics cannot be factored further. To wrap up, the original trinomial is completely factored as \(x^2(4x^2 + 2x + 1)\). This is the simplest form achievable over the real numbers.