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.

Case Study 1: Classifying Polynomial Expressions

Problem: 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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Case Study 2: Identifying Polynomial Characteristics

Problem: 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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Example 1: Even Degree, Positive Leading Coefficient (Basic)

Problem: Determine the end behavior of f(x)=2x4−3x2+1f(x) = 2x^4 - 3x^2 + 1.

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Example 2: Odd Degree, Negative Leading Coefficient (Intermediate)

Problem: Determine the end behavior of g(x)=−5x3+x2−4xg(x) = -5x^3 + x^2 - 4x.

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Example 3: Factored Form End Behavior (Advanced)

Problem: 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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Example 1: Polynomial Long Division (Intermediate)

Problem: Divide (2x3−5x2+3x−4)(2x^3 - 5x^2 + 3x - 4) by (x−2)(x - 2).

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Example 2: Synthetic Division (Intermediate)

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

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Example 1: Rational Root Theorem (Advanced)

Problem: Find all roots of P(x)=x3−4x2+x+6P(x) = x^3 - 4x^2 + x + 6.

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Example 1: Evaluating a Polynomial (Basic)

Problem: 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 2: The Factor Theorem (Intermediate)

Problem: 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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