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Adding and Subtracting Polynomials

Adding and Subtracting Polynomials

Adding and Subtracting Polynomials

Introduction

In this video, Professor Dave explains the basics of adding and subtracting polynomials. The video focuses on combining like terms and using the distributive property to simplify polynomial expressions.

Main Points

- Polynomials are expressions with multiple terms, each including a variable raised to a positive whole number exponent, possibly multiplied by a coefficient.

- Terms are listed in descending order of exponents, ending with a constant term.

- To add or subtract polynomials, combine like terms, which are terms with the same variable raised to the same exponent.

- Example: (5x3 + 4x3 = (5 + 4)x3 = 9x3).

- Example 1: (4x3 + 2x2 - 6x + 9) and (x4 - 3x2 + 6x - 4).

- Align the terms by their degrees and add:

    \[
    \begin{aligned}
    &x^4 \\
    + &4x^3 \\
    + &(2x^2 - 3x^2) = -x^2 \\
    + &(-6x + 6x) = 0 \\
    + &(9 - 4) = 5 \\
    \end{aligned}
    \]
    - Result: \(x^4 + 4x^3 - x^2 + 5\).

- Example 2: Subtract (x4 - 3x2 + 6x - 4) from (4x3 + 2x2 - 6x + 9).

- Subtracting involves changing the signs of the terms being subtracted:

    \[
    \begin{aligned}
    &0 - x^4 = -x^4 \\
    &4x^3 - 0 = 4x^3 \\
    &2x^2 - (-3x^2) = 2x^2 + 3x^2 = 5x^2 \\
    &-6x - 6x = -12x \\
    &9 - (-4) = 9 + 4 = 13 \\
    \end{aligned}
    \]
    - Result: \(-x^4 + 4x^3 + 5x^2 - 12x + 13\).
Conclusion

Adding and subtracting polynomials involves combining like terms and carefully managing signs during subtraction. This foundational skill is essential before moving on to more complex operations like multiplying polynomials.

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