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Number and Algebra : Module 39 Years : PDF Version of module. Polynomials represent the next level of algebraic complexity after quadratics.
Indeed a quadratic is a polynomial of degree 2. We can factor quadratic expressions, solve quadratic equations and graph quadratic functions, the obvious question arises as to how these things might be performed with algebraic expressions of higher degree.
In this module we will see how to arrive at this factorisation. Polynomials in many respects behave like whole numbers or the integers. We can add, subtract and multiply two or more polynomials together to obtain another polynomial.
Just as we can divide one whole number by another, producing a quotient and remainder, we can divide one polynomial by another and obtain a quotient and remainder, which are also polynomials. Similarly, information about the roots of a polynomial equation enables us to give a rough sketch of the corresponding polynomial function.
As well as being intrinsically interesting objects, polynomials have important applications in the real world. One such application to error-correcting codes is discussed in the Appendix to this module. There may be any number of terms, but each term must be a multiple of a whole number power of x. The term with the highest power of is called the leading term and its coefficient is called the leading coefficicent.
If the leading coefficient is 1, then the polynomial is called monic. The index of the leading term is called the degree of the polynomial. The term independent of is called the constant term. In the first polynomial, the coefficients are all integer while the second polynomials has an irrational coefficient.
For the most part, we will consider only polynomials of the first type, but much of what follows applies equally well to the second. To name polynomials, we will use the function notation such as p x or q x. This enables us to conveniently substitute values of x when required.
The coefficients are, in general, real numbers. Write down the leading term, the leading coefficient, the degree and the constant term in the general polynomial given above. A polynomial that consists only of a non-zero constant, is called a constant polynomial and has degree 0. It has no terms and so there is no leading term. It is best not to define the degree of the zero polynomial. Adding, subtracting and multiplying polynomials.
Notice that we usually write the terms in the polynomial from largest to smallest degree. This is sometimes called the standard form of the polynomial. To multiply two polynomials, we multiply each term in the first polynomial by second polynomial and collect like terms. When one whole number is divided into another a quotient and remainder is formed. Thus gives 7 remainder 2. There are various ways to write this result. The key point about the remainder is that it is non-negative but strictly less than the divisor.
If the remainder is zero, then we cay that d is a factor of p. These basic statements regarding arithmetic have analogues when we come to divide one polynomial by another.
We will use the same terminology when discussing polynomial division. The process of doing this is modelled on long division. The division of the polynomial p x by the polynomial d x also produces a quotient q x and a remainder r x and so we can write.
The key idea in performing the division is to keep working with the leading terms, as the following example shows. It can be seen from the above example that the degree of the remainder is less than the degree of the divisor, since otherwise, we could continue the division.
Thus, in the case when is a linear factor, the remainder will be a constant and so we can write it as. Note that in this case, since the divisor has degree 2, the remainder will either be 0 or have degree at most 1.
Long division of polynomials is a cumbersome process and in some instances we are only interested in the remainder. This does not appear until the end of the computation.
This surprising result is called the remainder theorem. Find the value of the coefficient a. Factoring quadratics is an important technique which we used to solve quadratic equations. In a similar way, we would like to be able to develop some techniques to factor polynomials. Using the remainder theorem, we have now proven:. We could thus find the third factor by long division. Where a is a number to be determined.
Find the values of the coefficients a and b. Our aim is to take a polynomial with integer coefficients and write it as a product of polynomials of smaller degree which also has integer coefficients.
This process is called factoring over the integers. Thus we may be able to repeat the process on q x and so on as often as possible to give a complete factorisation of p x. To assist us in finding an integer zero of the polynomial we use the following result. One of the main methods of solving quadratic equations was the method of factoring.
Similarly, one of the main applications of factoring polynomials is to solve polynomial equations. Since the product of the three factors is zero, we can equate each factor to zero to find the solutions.
Thus the number of distinct solutions may be less than the degree, but it can never e x ceed the degree. In some situations, the factorisation results in a quadratic equation with either no real solutions or irrational solutions.
In this case, we may need to complete the problem by using the quadratic formula. Note that there are polynomial equations with irrational roots that cannot be solved using the procedure above. In general, factoring polynomials over the integers is a difficult problem. In the module, Quadratic Functions we saw how to sketch the graph of a quadratic by locating.
The verte x is an e x ample of a turning point. For polynomials of degree greater than 2, finding turning points is not an elementary procedure and usually requires the use of calculus, however:.
To get a picture of the overall shape of the curve, we can substitute some test points. We can represent the sign of y using a sign diagram:. It does not tell us the ma x imum and minimum values of y between the zeroes. Notice that if x is a large positive number, then p x is also large and positive.
If x is a large negative number, then p x is also a large negative number. If we e x amine, for e x ample, the size of x 4 for various values of x , we notice. In the case of the parabola, we call this a verte x but we do not generally use this word for polynomials of higher degree. Instead we talk of a turning point and further classify it as a ma x imum or minimum.
In the following we will consider odd powers greater or equal to 3. As above, the graph is flat near the origin. At the origin we have neither a ma x imum nor a minimum. The sign diagram is. The zeroes of a polynomial are also called the roots of the corresponding polynomial equation. To properly understand how many solutions a polynomial equation may have, we need to introduce the comple x numbers. The comple x number i is often referred to as an imaginary number.
Every polynomial equation of degree greater than 0, has at least one comple x solution. Every polynomial equation of degree n , greater than 0, has e x actly n solutions, counting multiplicity, over the comple x numbers. E x plain how the corollary may be deduced from the theorem. Hence, every polynomial of degree n , greater than 0, can be factored into n linear factors using comple x numbers.
However, the equation has only two distinct roots. The verte x of a parabola is an e x ample of a turning point. The x -coordinates of the turning points of a polynomial are not so easy to find and require the use of differential calculus which is studied in senior mathematics.
We can perform a similar e x ercise on monic cubics. These identities give relationships between the roots of a polynomial and its coefficients.
The study of equations of degree greater than two goes back to Arabic mathematics. It was not until the Renaissance that the general solution of the cubic was obtained.
We now e x pand the left-hand side and factor 3uv from two of the terms to give. At this stage, we have two numbers u 3 , v 3 whose sum and product we know. Use your calculator to e x press this in decimal form and check that it satisfies the original equation. He discovered a method to reduce the problem of solving a quartic to that of solving a cubic.
In both cases it is possible to e x press the solution of the given equation using square and higher roots and the usual operations of arithmetic addition, subtraction, multiplication and division.
A monomial is a number, a variable or a product of a number and a variable where all exponents are whole numbers. That means that. The degree of the monomial is the sum of the exponents of all included variables. Constants have the monomial degree of 0. A polynomial as oppose to the monomial is a sum of monomials where each monomial is called a term. The degree of the polynomial is the greatest degree of its terms. A polynomial is usually written with the term with the highest exponent of the variable first and then decreasing from left to right.
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Skip to main content. Search form Search. Polynomial in standard form. Polynomial in standard form polynomial in standard form 75x x4 b. The In order to write any polynomial in standard form, you look at the degree of each term. If Power is not a whole number , the equation is not a polynomial.
Number and Algebra : Module 39 Years : PDF Version of module.
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Worksheet by Kuta Software LLC. Kuta Software - Infinite Naming Polynomials. Name each polynomial by degree and number of terms. 1) 2p. 4 + p. 3. 2) −.Edita A. 20.05.2021 at 19:50
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Elementary Algebra Skill. Naming Polynomials. Name each polynomial by degree and number of terms. 1) 3. 2) 6a. 3. 3) 9m + 5m. 2 + 10m. 3 − 5. 4) −9x + 4x.Katrin F. 26.05.2021 at 08:57
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