Combining Like Terms in Polynomials
Welcome back to Professor Baker's Math Class! Today, we dove into the world of polynomials and learned how to simplify them by combining like terms. This is a crucial skill for success in algebra, and we're here to help you master it.
What are Like Terms?
Like terms are terms that have the same variable(s) raised to the same power(s). For example, $3x^2$ and $6x^2$ are like terms because they both have the variable $x$ raised to the power of 2. However, $7x$ and $8x^2$ are not like terms because they have different powers of $x$. Remember, you can only combine terms when the variables and exponents are exactly the same!
How to Combine Like Terms:
- Identify Like Terms: Look for terms with the same variable and exponent.
- Rearrange (Optional): You can rearrange the expression to group like terms together. This can make it easier to combine them.
- Combine Coefficients: Add or subtract the coefficients of the like terms. The variable and exponent stay the same.
Example 1:
Simplify the following expression: $3x^2 - 5 + 7x + 6x^2$
First, identify the like terms: $3x^2$ and $6x^2$ are like terms. So, the simplified form is:
$f(x) = (3x^2 + 6x^2) + 7x - 5 = 9x^2 + 7x - 5$.
This resulting expression is a quadratic trinomial with a leading coefficient of 9.
Example 2:
Simplify the expression: $6x^3 - 7x^3 - 4x^2 - 9$
Identify the like terms: $6x^3$ and $-7x^3$. Now combine:
$f(x) = (6x^3 - 7x^3) - 4x^2 - 9 = -x^3-4x^2 - 9$.
Example 3:
Simplify: $16x - 3x + 4y - 2y + 2x$
Group and combine: $(16x - 3x + 2x) + (4y - 2y) = 15x + 2y$
Polynomial Vocabulary
- Monomial: An expression with one term (e.g., $5x$, $7$).
- Binomial: An expression with two terms (e.g., $x + 2$, $3y - 1$).
- Trinomial: An expression with three terms (e.g., $x^2 + 2x + 1$, $a + b - c$).
- Polynomial: A general term for an expression with one or more terms.
- Degree of a Term: The exponent of the variable in the term.
- Leading Coefficient: The coefficient of the term with the highest degree when the polynomial is written in standard form (highest to lowest power).
Homework: Don't forget to complete the packet for more practice. Keep practicing, and you'll become a pro at combining like terms!