Examples of Leading coefficient in the following topics:
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- anxn 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 −14x4 as its leading term and −141 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−x0)k equals x0k for all positive integers k, we can substitute x by t+x0 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 x0.
- 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)=ax2+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 x2 term is given by the product of the coefficients α1 and α2, 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 axbyc 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 ai 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 yn 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 axbyc, 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 axbyc is known as the binomial coefficient nb or nc (the two have the same value).
- These coefficients for varying n and b can be arranged to form Pascal's triangle.