X3 2x2 6 is what kind of polynomial

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. A polynomial is basically a string of mathematical clumps called terms all added together. Each individual clump usually consists of one or more variables raised to exponential powers, usually with a coefficient attached. Polynomials are usually written in standard form, which means that the terms are listed in order from the largest exponential value to the term with the smallest exponent. Because the term containing the variable raised to the highest power is listed first in standard form, its coefficient is called the leading coefficient.

A polynomial not containing a variable is called the constant. A polynomial consists of the sum of distinct algebraic clumps called terms , each of which consists of a number, one or more variables raised to an exponent, or both.

The largest exponent in the polynomial is called the degree , and the coefficient of the variable raised to that exponent is called the leading coefficient. The constant in a polynomial has no variable written next to it.

Note that each term's variable has a lower power than the term to its immediate left. The degree of this polynomial is 5, its leading coefficient is -7, and the constant is 1.

Remember coefficients have nothing at all do to with the degree. The answer is 3 since the that is the largest exponent. The answer is 3. Remember ignore those coefficients. The answer is 9. The answer is 8. Be careful sometimes polynomials are not ordered from greatest exponent to least. Even though 7x 3 is the first expression, its exponent does not have the greatest value.

The answer is The answer is 2. Do NOT count any constants "constant" is just a fancy math word for 'number'.