Examples of Leading coefficient in the following topics:
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- $a_nx^n$ 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:
- which has $-\frac {x^4}{14}$ as its leading term and $- \frac{1}{14}$ as its leading coefficient.
- As the degree is even and the leading coefficient is negative, the function declines both to the left and to the right.
- Because the degree is odd and the leading coefficient is positive, the function declines to the left and inclines to the right.
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- Synthetic division only works for polynomials divided by linear expressions with a leading coefficient equal to $1.$
- We start by writing down the coefficients from the dividend and the negative second coefficient of the divisor.
- Bring down the first coefficient and multiply it by the divisor.
- Then add the next column of coefficients, get the result and multiply that by the divisor to find the third coefficient $-27$:
- Thus $1$ is a zero of a polynomial if and only if its coefficients add to $0.$
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- When given a polynomial with integer coefficients, we can plug in all of these candidates and see whether they are a zero of the given polynomial.
- Since every polynomial with rational coefficients can be multiplied with an integer to become a polynomial with integer coefficients and the same zeros, the Rational Root Test can also be applied for polynomials with rational coefficients.
- 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 substitute $x$ with $t+1$ and obtain a polynomial in $t$ with leading coefficient $3$ and constant term $1$.
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- The discriminant of a polynomial is a function of its coefficients that reveals information about the polynomial's roots.
- The discriminant of a quadratic function is a function of its coefficients that reveals information about its roots.
- Where $a$, $b$, and $c$ are the coefficients in $f(x) = ax^2 + bx + c$.
- Since adding zero and subtracting zero in the quadratic equation lead to the same outcome, there is only one distinct root of the quadratic function.
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- 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:
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- Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph.
- The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex.
- If the coefficient $a>0$, the parabola opens upward, and if the coefficient $a<0$, the parabola opens downward.
- The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex.
- The coefficient $c$ controls the height of the parabola.
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- Polynomials with rational coefficients should be treated and worked the same as other polynomials.
- Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals".
- However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.
- Polynomials with rational coefficients can be treated just like any other polynomial, just remember to utilize all the properties of fractions necessary during your operations.
- Extend the techniques of finding zeros to polynomials with rational coefficients
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- A coefficient is a numerical value which multiplies a variable (the operator is omitted).
- When a coefficient is one, it is usually omitted.
- Added terms are simplified using coefficients.
- For example, $x+x+x$ can be simplified as $3x$ (where 3 is the coefficient).
- 1 – Exponent (power), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, x,y – variables
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- The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
- Any coefficient $a$ in a term $ax^by^c$ of the expanded version is known as a binomial coefficient.
- Notice the coefficients are the numbers in row two of Pascal's triangle: $1,2,1$.
- Where the coefficients $a_i$ in this expansion are precisely the numbers on row $n$ of Pascal's triangle.
- Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of $y^n$ in these binomial expansions, while the next diagonal corresponds to the coefficient of $xy^{n−1}$ and so on.
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- The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.
- According to the theorem, it is possible to expand the power $(x + y)^n$ into a sum involving terms of the form $ax^by^c$, where the exponents $b$ and $c$ are nonnegative integers with $b+c=n$, and the coefficient $a$ of each term is a specific positive integer depending on $n$ and $b$.
- The coefficient a in the term of $ax^by^c$ is known as the binomial coefficient $n^b$ or $n^c$ (the two have the same value).
- These coefficients for varying $n$ and $b$ can be arranged to form Pascal's triangle.