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(2x^2+x^3+1) (x^2+1)

3 min read Jun 16, 2024
(2x^2+x^3+1) (x^2+1)

Multiplying Polynomials: (2x^2 + x^3 + 1)(x^2 + 1)

This article will explore how to multiply the two polynomials: (2x^2 + x^3 + 1) and (x^2 + 1). We will use the distributive property to achieve this multiplication.

The Distributive Property

The distributive property states that for any numbers a, b, and c:

a(b + c) = ab + ac

We can extend this property to multiplying polynomials by distributing each term of one polynomial across all terms of the other polynomial.

Multiplying the Polynomials

  1. Distribute the first term of the first polynomial (2x^2):

    (2x^2)(x^2 + 1) = 2x^4 + 2x^2

  2. Distribute the second term of the first polynomial (x^3):

    (x^3)(x^2 + 1) = x^5 + x^3

  3. Distribute the third term of the first polynomial (1):

    (1)(x^2 + 1) = x^2 + 1

  4. Combine all the terms:

    2x^4 + 2x^2 + x^5 + x^3 + x^2 + 1

  5. Rearrange the terms in descending order of exponents:

    x^5 + 2x^4 + x^3 + 3x^2 + 1

Therefore, the product of (2x^2 + x^3 + 1) and (x^2 + 1) is x^5 + 2x^4 + x^3 + 3x^2 + 1.

Key Points

  • The distributive property is a fundamental tool for multiplying polynomials.
  • It's important to distribute each term of one polynomial across all terms of the other polynomial.
  • Combine like terms after distributing to simplify the expression.
  • Rearrange the terms in descending order of exponents for standard polynomial form.

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