Polynomials - Examples & Applications

This section provides step-by-step examples on how to perform operations with polynomials. You will learn to classify polynomials, understand their end behavior, perform long and synthetic division, and apply the Remainder Theorem.

Classifying Polynomials

Polynomials are classified by their degree (the highest exponent) and their number of terms.

Example

Case Study 1: Classifying Polynomial Expressions Classify the following polynomials by their degree and number of terms:
  1. 5x2−3x+25x^2 - 3x + 2
  2. −7x4+x-7x^4 + x
  3. 1212
  4. 2x3−x2+4x−12x^3 - x^2 + 4x - 1

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Example

Case Study 2: Identifying Polynomial Characteristics For the polynomial P(x)=−3x5+2x3−7x+4P(x) = -3x^5 + 2x^3 - 7x + 4, identify the leading term, leading coefficient, degree, and constant term.

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End Behavior

The end behavior of a polynomial graph describes what happens to the function value f(x)f(x) as xx approaches positive or negative infinity. It is determined by the leading term.

Example

Example 1: Even Degree, Positive Leading Coefficient (Basic) Determine the end behavior of f(x)=2x4−3x2+1f(x) = 2x^4 - 3x^2 + 1.

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Example

Example 2: Odd Degree, Negative Leading Coefficient (Intermediate) Determine the end behavior of g(x)=−5x3+x2−4xg(x) = -5x^3 + x^2 - 4x.

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Example

Example 3: Factored Form End Behavior (Advanced) Determine the end behavior of h(x)=−2(x−1)2(x+3)(x−4)2h(x) = -2(x - 1)^2(x + 3)(x - 4)^2.

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Division and Factorization

Polynomial division is used to simplify rational expressions and find roots. Synthetic division is a shortcut for dividing by a linear binomial (x−c)(x - c).

Example

Example 1: Polynomial Long Division (Intermediate) Divide (2x3−5x2+3x−4)(2x^3 - 5x^2 + 3x - 4) by (x−2)(x - 2).

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Example

Example 2: Synthetic Division (Intermediate) Use synthetic division to divide (3x3−2x2+5)(3x^3 - 2x^2 + 5) by (x+1)(x + 1).

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Finding Roots

Finding roots involves setting the polynomial equal to zero. The Rational Root Theorem and synthetic division are key tools for higher-degree polynomials.

Example

Example 1: Rational Root Theorem (Advanced) Find all roots of P(x)=x3−4x2+x+6P(x) = x^3 - 4x^2 + x + 6.

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The Remainder Theorem

The Remainder Theorem states that if a polynomial P(x)P(x) is divided by (x−c)(x - c), the remainder is equal to P(c)P(c). This provides a fast way to evaluate polynomials.

Example

Example 1: Evaluating a Polynomial (Basic) Let P(x)=x4−3x2+5x−2P(x) = x^4 - 3x^2 + 5x - 2. Find the remainder when P(x)P(x) is divided by (x−2)(x - 2).

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Example

Example 2: The Factor Theorem (Intermediate) Determine if (x+3)(x + 3) is a factor of P(x)=2x3+5x2−4x−3P(x) = 2x^3 + 5x^2 - 4x - 3.

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