Factor completely, or state that the polynomial is prime. $x^2-11x+28$

Verified step by step guidance

1. Identify the quadratic polynomial to factor: \(x^2 - 11x + 28\).
2. Recall that to factor a quadratic of the form \(x^2 + bx + c\), we look for two numbers that multiply to \(c\) and add to \(b\).
3. Find two numbers that multiply to \$28\( and add to \)-11\(. Consider the factors of \)28\(: \)1 \times 28\(, \)2 \times 14\(, \)4 \times 7$.
4. Since the sum is negative and the product is positive, both numbers must be negative. Check if \(-4\) and \(-7\) work: \(-4 \times -7 = 28\) and \(-4 + (-7) = -11\).
5. Write the factored form using these numbers: \((x - 4)(x - 7)\).

Key Concepts

Factoring Quadratic Polynomials

Factoring quadratic polynomials involves expressing a quadratic expression as a product of two binomials. For a quadratic in the form ax^2 + bx + c, we look for two numbers that multiply to ac and add to b. This process simplifies solving equations and analyzing polynomial properties.

Prime Polynomials

A prime polynomial is one that cannot be factored into polynomials of lower degree with coefficients in the given number system. Recognizing when a quadratic is prime helps determine if factoring is possible or if the polynomial must be left as is.

The Relationship Between Coefficients and Roots

The coefficients of a quadratic polynomial relate directly to its roots through the sum and product: the sum of the roots equals -b/a, and the product equals c/a. Understanding this helps identify factor pairs quickly when factoring quadratics.