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How to Distribute Polynomials?

Published in Polynomial Algebra 3 mins read

Distributing polynomials involves multiplying each term of one polynomial by each term of the other polynomial. This process relies on the distributive property and combining like terms.

Here's a breakdown of how to distribute polynomials:

1. Understanding the Distributive Property:

The distributive property states that a(b + c) = ab + ac. This principle is fundamental to polynomial distribution. We extend this when multiplying polynomials with multiple terms.

2. Multiplying a Monomial by a Polynomial:

This is the simplest case. Multiply the monomial by each term inside the polynomial.

  • Example: 3x(2x2 + 5x - 1) = (3x 2x2) + (3x 5x) + (3x * -1) = 6x3 + 15x2 - 3x

3. Multiplying a Binomial by a Binomial (FOIL Method):

The FOIL method is a mnemonic device for multiplying two binomials:

  • First: Multiply the first terms of each binomial.

  • Outer: Multiply the outer terms of each binomial.

  • Inner: Multiply the inner terms of each binomial.

  • Last: Multiply the last terms of each binomial.

  • Example: (x + 2)(x + 3) = (x x) + (x 3) + (2 x) + (2 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6

4. Multiplying Polynomials with Multiple Terms:

For polynomials with more than two terms, systematically multiply each term in the first polynomial by each term in the second polynomial.

  • Example: (x + 2)(x2 + 3x - 4)
    • Multiply 'x' by each term in the second polynomial: x(x2 + 3x - 4) = x3 + 3x2 - 4x
    • Multiply '2' by each term in the second polynomial: 2(x2 + 3x - 4) = 2x2 + 6x - 8
    • Combine the results: (x3 + 3x2 - 4x) + (2x2 + 6x - 8) = x3 + 5x2 + 2x - 8

5. Combining Like Terms:

After multiplying, simplify the resulting expression by combining like terms (terms with the same variable and exponent). As illustrated in the example from the YouTube video, combining like terms is crucial for obtaining the final simplified polynomial. For example: -3x2 - 6x2 = -9x2.

Key Takeaways:

  • Systematic Approach: Ensure you multiply every term in the first polynomial by every term in the second polynomial.
  • Careful with Signs: Pay close attention to positive and negative signs during multiplication.
  • Combine Like Terms: Simplify the expression by combining terms with the same variable and exponent.

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