Adding and subtracting polynomials involves combining like terms. Here's a step-by-step guide:
1. Standard Form Arrangement
First, arrange each polynomial in its standard form. This means writing the terms in order of decreasing exponent. For example, 3x^2 + 5x - 2 is in standard form.
2. Identifying Like Terms
Like terms are terms that have the same variable raised to the same power. For instance, 3x^2 and -x^2 are like terms, while 3x^2 and 3x are not like terms.
3. Addition of Polynomials
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Horizontally: Write the polynomials next to each other, enclosed in parentheses and separated by a plus sign.
(Polynomial 1) + (Polynomial 2)
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Remove Parentheses: Because it's addition, the signs inside the parentheses will remain the same. Remove the parentheses.
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Combine Like Terms: Identify and combine like terms by adding their coefficients. Remember that coefficients are the numerical part of the term (e.g., in
3x^2, the coefficient is 3).Example:
(2x^2 + 3x - 1) + (x^2 - 4x + 5)
= 2x^2 + 3x - 1 + x^2 - 4x + 5
= (2x^2 + x^2) + (3x - 4x) + (-1 + 5)
= 3x^2 - x + 4
4. Subtraction of Polynomials
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Horizontally: Write the polynomials next to each other, enclosed in parentheses and separated by a minus sign.
(Polynomial 1) - (Polynomial 2)
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Distribute the Negative Sign: This is crucial! The minus sign in front of the second polynomial means you need to distribute it to every term inside the parentheses. Change the sign of each term in the second polynomial.
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Remove Parentheses: After distributing the negative sign, you can remove the parentheses.
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Combine Like Terms: Identify and combine like terms by adding their coefficients.
Example:
(2x^2 + 3x - 1) - (x^2 - 4x + 5)
= 2x^2 + 3x - 1 - x^2 + 4x - 5(Notice the signs changed in the second polynomial)
= (2x^2 - x^2) + (3x + 4x) + (-1 - 5)
= x^2 + 7x - 6
5. Organizing the Result
After combining like terms, write the resulting polynomial in standard form (decreasing order of exponents).
Example with Multiple Variables:
Add: (3x^2y - 2xy + 5y^2) + (x^2y + 5xy - 2y^2)
= 3x^2y - 2xy + 5y^2 + x^2y + 5xy - 2y^2
= (3x^2y + x^2y) + (-2xy + 5xy) + (5y^2 - 2y^2)
= 4x^2y + 3xy + 3y^2
