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

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Identify the quadratic polynomial to factor: $x^2 - 11x + 28$.

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 $b$.

Find two numbers that multiply to $28$ and add to $-11$. Consider the factors of $28$: $1 \times 28$, $2 \times 14$, $4 \times 7$.

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 $-4 + (-7) = -11$.

Write the factored form using these numbers: $(x - 4)(x - 7)$.

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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.

Factor completely, or state that the polynomial is prime. x2−11x+28x^2-11x+28