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Understanding 'like terms' is crucial when working with polynomials. In algebra, like terms are terms that have the same variables raised to the same powers. For example, in the expression \(15x^4 + 10x^3\), the terms \(15x^4\) and \(-6x^4\) are like terms because they both contain \(x^4\). Similarly, \(10x^3\) and \(9x^3\) are like terms because they both contain \(x^3\).
Like terms can be combined to simplify expressions and polynomials. This is because their variables and exponents match, making it possible to add or subtract their coefficients to form a single term.
For example:

• \(2x^2 + 3x^2 = 5x^2\)
• \(7x - 5x = 2x\)

Identifying like terms is the first step in many algebraic operations, such as adding and subtracting polynomials.

Polynomial coefficients are the numerical parts of the terms in a polynomial. They precede the variables and their exponents. For instance, in the polynomial \(15x^4 + 10x^3\), the numbers 15 and 10 are the coefficients. When combining polynomials, these coefficients are added or subtracted based on their corresponding like terms.
Coefficients can be positive or negative, and they play a key role in determining the value of the polynomial. Here are few points to note about coefficients:

• Always look at the sign (+ or -) in front of the coefficient to determine whether to add or subtract when combining terms.
• When like terms have opposite signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the larger absolute value.

Understanding how to work with coefficients is essential for correctly manipulating and simplifying polynomials.

Combining like terms involves adding or subtracting the coefficients of like terms to create a simplified polynomial. This step is fundamental in polynomial addition and subtraction. To combine like terms, follow these steps:
1. **Identify the like terms**: Look for terms in the polynomials that have the same variable and exponent. For example, in \(15x^4 + 10x^3\) and \(-6x^4 + 9x^3\), \(15x^4\) and \(-6x^4\) are like terms.
2. **Combine the coefficients**: Add or subtract the coefficients of the like terms. With \(15x^4\) and \(-6x^4\), you add 15 and -6 to get 9. Similarly, for \(10x^3\) and \(9x^3\), you add 10 and 9 to get 19.
3. **Write the new term**: Combine the new coefficients with their corresponding variables and exponents. The resulting polynomial from our example would be \(9x^4 + 19x^3\).
Breaking down the process into these steps can help you accurately and efficiently combine like terms in polynomials.