Leading term

Examples of Leading term in the following topics:

• The Leading-Term Test

  • $a_nx^n$ is called the leading term of $f(x)$, while $a_n \not = 0$ is known as the leading coefficient.
  • The properties of the leading term and leading coefficient indicate whether $f(x)$ increases or decreases continually as the $x$-values approach positive and negative infinity:
  • In the leading term, $a_n$ equals $\frac {1}{4}$ and $n$ equals $3$.
  • which has $-\frac {x^4}{14}$ as its leading term and $- \frac{1}{14}$ 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 $(x-x_0)^k$ equals $x_0^k$ for all positive integers $k$, we can substitute $x$ by $t+x_0$ 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 $x_0$.
  • In this case we substitue $x$ by $t+1$ and obtain a polynomial in $t$ with leading coefficient $3$ and constant term $1$.

• 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 $x^2$ term is given by the product of the coefficients $\alpha_1$ and $\alpha_2$, and the coefficient of the $x$ 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 $1.$
  • Note that we explicitly write out all zero terms!

• Adding and Subtracting Polynomials

  • For example, $4x^3$ and $x^3$are like terms; $21$ and $82$ 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 $x^2$, 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.