Title: How to Use the Remainder Theorem to Evaluate a Polynomial

This post shows how to apply the Remainder Theorem to evaluate a polynomial without plugging directly into the equation. It's a key concept in precalculus and early calculus that students often overlook—but it can save you time and help on exams.

The Problem:
Given
P(x) = 2x³ – 9x² – 7x + 13
Find P(5) using the Remainder Theorem.

What the Remainder Theorem Says:
If a polynomial P(x) is divided by (x – a), the remainder is P(a).
So to find P(5), divide P(x) by (x – 5) using long division or synthetic division.

Step-by-Step Breakdown (Using Long Division):

• Divide 2x³ – 9x² – 7x + 13 by x – 5

• The quotient is: 2x² + x – 2

• The remainder is: 3

• Therefore: P(5) = 3

The visual shows both long division and how to substitute back using:
P(x) = (x – a)(quotient) + remainder
Then plug in x = 5 to verify.

Key Insight:
Knowing how to divide polynomials and apply this theorem is essential for calculus topics like partial fractions and limits. Many students lose points not because of calculus, but because they never practiced factoring or division properly in algebra.

This walkthrough comes straight from Chapter 4 of the Ultimate Crash Course for STEM Majors, designed to help you finally master all foundational math tools for higher-level work.

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