coefficient
a constant by which an algebraic term is multiplied.
A quantity (usually a number) that remains the same in value within a problem.
Examples of coefficient in the following topics:
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Graphing Quadratic Equations In Standard Form
- Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph.
- The coefficient controls the speed of increase (or decrease) of the quadratic function from the vertex.
- If the coefficient , the parabola opens upward, and if the coefficient , the parabola opens downward.
- The coefficients and together control the axis of symmetry of the parabola and the -coordinate of the vertex.
- The coefficient controls the height of the parabola.
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Zeroes of Polynomial Functions With Rational Coefficients
- 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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The Remainder Theorem and Synthetic Division
- 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 :
- A special case of this is when the left number is : then the last number equals the sum of all coefficients!
- Thus is a zero of a polynomial if and only if its coefficients add to
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Simplifying Algebraic Expressions
- 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, can be simplified as (where 3 is the coefficient).
- 1 – Exponent (power), 2 – Coefficient, 3 – term, 4 – operator, 5 – constant, x,y – variables
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Binomial Expansions and Pascal's Triangle
- The binomial theorem, which uses Pascal's triangles to determine coefficients, describes the algebraic expansion of powers of a binomial.
- Any coefficient in a term of the expanded version is known as a binomial coefficient.
- Notice the coefficients are the numbers in row two of Pascal's triangle: .
- Where the coefficients in this expansion are precisely the numbers on row of Pascal's triangle.
- Notice that the entire right diagonal of Pascal's triangle corresponds to the coefficient of in these binomial expansions, while the next diagonal corresponds to the coefficient of $xy^{n−1}$ and so on.
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Total Number of Subsets
- 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 into a sum involving terms of the form , where the exponents and are nonnegative integers with , and the coefficient of each term is a specific positive integer depending on and .
- The coefficient a in the term of is known as the binomial coefficient or (the two have the same value).
- These coefficients for varying and can be arranged to form Pascal's triangle.
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The Fundamental Theorem of Algebra
- The fundamental theorem states that every non-constant, single-variable polynomial with complex coefficients has at least one complex root.
- Some polynomials with real coefficients, like , have no real zeros.
- As it turns out, every polynomial with a complex coefficient has a complex zero.
- Every polynomial of odd degree with real coefficients has a real zero.
- In particular, every polynomial of odd degree with real coefficients admits at least one real root
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Integer Coefficients and the Rational Zeros Theorem
- 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 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 substitute with and obtain a polynomial in with leading coefficient and constant term .
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The Leading-Term Test
- is called the leading term of , while $a_n \not = 0$ is known as the leading coefficient.
- which has as its leading term and as its leading coefficient.
- and the absolute value of is bigger than , where is the absolute value of the largest coefficient divided by the leading coefficient, is the degree of the polynomial and is a big number, then the absolute value of will be bigger than times the absolute value of any other term, and bigger than times the other terms combined!
- 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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Matrix Equations
- Make sure that one side of the equation is only variables and their coefficients, and the other side is just constants.
- To solve a system of linear equations using an inverse matrix, let be the coefficient matrix, let be the variable matrix, and let be the constant matrix.
- If the coefficient matrix is not invertible, the system could be inconsistent and have no solution, or be dependent and have infinitely many solutions.