Adding and Subtracting Polynomials

Welcome to today's lesson where we'll be diving into the world of polynomials! We're going to learn how to add and subtract these expressions, which is a fundamental skill in algebra. Remember, the key is to combine like terms. Let's get started!

What are Polynomials?

A polynomial is an expression containing variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, $3x^2 + 2x - 5$ is a polynomial.

Adding Polynomials

To add polynomials, you can use either a vertical or horizontal format. The most important thing is to combine like terms. Like terms are terms that have the same variable raised to the same power.

Example (Horizontal Method):

Let's add $(4m^2 + 5) + (m^2 - m + 6)$.

First, remove the parentheses: $4m^2 + 5 + m^2 - m + 6$

Then, combine like terms: $(4m^2 + m^2) - m + (5 + 6)$

Finally, simplify: $5m^2 - m + 11$

Example (Vertical Method):

To add vertically, align like terms in columns and add:

$$ \begin{array}{c@{\,}c@{\,}c@{\,}c} & 5x^2 & +4x & +1 \\ + & 2x^2 & +5x & +2 \\ \hline & 7x^2 & +9x & +3 \end{array} $$

Subtracting Polynomials

Subtracting polynomials is similar to adding, but you need to remember to distribute the negative sign to each term in the polynomial being subtracted. In essence, subtracting is the same as adding the opposite.

Example:

Let's subtract $(x^3 + 4y) - (2x^3)$.

First, distribute the negative sign: $x^3 + 4y - 2x^3$

Then, combine like terms: $(x^3 - 2x^3) + 4y$

Finally, simplify: $-x^3 + 4y$

Another Example:

Subtract $(7m^4 - 2m^2) - (5m^4 - 5m^2 + 8)$

Distribute the negative: $7m^4 - 2m^2 - 5m^4 + 5m^2 - 8$

Combine like terms: $(7m^4 - 5m^4) + (-2m^2 + 5m^2) - 8$

Simplify: $2m^4 + 3m^2 - 8$

Key Takeaways

  • Like Terms: Only combine terms with the same variable and exponent.
  • Distribute Negatives: When subtracting polynomials, remember to distribute the negative sign.
  • Organization: Use vertical or horizontal methods to keep your work organized.

Homework

Complete the worksheet to practice adding and subtracting polynomials. Pay close attention to signs and combining like terms. Keep up the great work!