Leading term
The term in a polynomial in which the independent variable is raised to the highest power.
Examples of Leading term in the following topics:
-
The Leading-Term Test
- is called the leading term of , while $a_n \not = 0$ is known as the leading coefficient.
- The properties of the leading term and leading coefficient indicate whether increases or decreases continually as the -values approach positive and negative infinity:
- In the leading term, equals and equals .
- which has as its leading term and as its leading coefficient.
- Use the leading-term test to describe the end behavior of a polynomial graph
-
Integer Coefficients and the Rational Zeroes Theorem
- Now we use a little trick: since the constant term of equals for all positive integers , we can substitute by to find a polynomial with the same leading coefficient as our original polynomial and a constant term equal to the value of the polynomial at .
- In this case we substitue by and obtain a polynomial in with leading coefficient and constant term .
-
Simplifying Matrices With Row Operations
- Before getting into more detail, there are a couple of key terms that should be mentioned:
- Using elementary row operations at the end of the first part (Gaussian elimination, zeros only under the leading 1) of the algorithm:
- At the end of the algorithm, if the Gauss–Jordan elimination (zeros under and above the leading 1) is applied:
-
Factoring General Quadratics
- In other words, the coefficient of the term is given by the product of the coefficients and , and the coefficient of the term is given by the inner and outer parts of the FOIL process.
- This leads to the factored form:
- This leads to the equation:
-
Direct Variation
- Any augmentation of one variable would lead to an equal augmentation of the other.
- Notice what happens when you change the "k" term.
-
The Remainder Theorem and Synthetic Division
- Synthetic division only works for polynomials divided by linear expressions with a leading coefficient equal to
- Note that we explicitly write out all zero terms!
-
Adding and Subtracting Polynomials
- For example, and are like terms; and are also like terms.
- When adding polynomials, the commutative property allows us to rearrange the terms to group like terms together.
- For example, one polynomial may have the term , while the other polynomial has no like term.
- If any term does not have a like term in the other polynomial, it does not need to be combined with any other term.
- Start by grouping like terms.
-
Adding and Subtracting Algebraic Expressions
- Terms are called like terms if they have the same variables and exponents.
- All constant terms are also like terms.
- Note that terms that share a variable but not an exponent are not like terms.
- Likewise, terms that share an exponent but have different variables are not like terms.
- When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together.
-
Sums, Differences, Products, and Quotients
- For instance, in the equation y = x + 5, there are two terms, while in the equation y = 2x2, there is only one term.
- We then collect like terms.
- A monomial equations has one term; a binomial has two terms; a trinomial has three terms.
- Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
- Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
-
Multiplying Algebraic Expressions
- A polynomial is called a binomial if it has two terms, and a trinomial if it has three terms.
- Any negative sign on a term should be included in the multiplication of that term.
- Outer ("outside" terms are multiplied—that is, the first term of the first binomial and the second term of the second)
- Inner ("inside" terms are multiplied—second term of the first binomial and first term of the second)
- Remember that any negative sign on a term in a binomial should also be included in the multiplication of that term.