Which binomial is a factor of x 3 x 2 3x 3

Five times the smaller integer. Five times the larger integer. Let us say we know the sum of two numbers is If we represent one number by x, then the second number must be 10 - x as suggested by the following table.

In general, if we know the sum of two numbers is 5 and x represents one number, the other number must be S - x. The next example concerns the notion of consecutive integers that was consid- ered in Section 3. The difference of the squares of two consecutive odd integers is The larger integer b. The square of the smaller integer c. The square of the larger integer. Sometimes, the mathematical models equations for word problems involve parentheses.

We can use the approach outlined on page to obtain the equation. Then, we proceed to solve the equation by first writing equivalently the equation without parentheses. One integer is five more than a second integer. Three times the smaller integer plus twice the larger equals Find the integers.

Steps First, we write what we want to find the integers as word phrases. Then, we represent the integers in terms of a variable. In this section, we will examine several applications of word problems that lead to equations that involve parentheses. Once again, we will follow the six steps out- lined on page when we solve the problems. The basic idea of problems involving coins or bills is that the value of a number of coins of the same denomination is equal to the product of the value of a single coin and the total number of coins.

There are 16 more dimes than quarters. How many dimes and quarters are in the col- lection? Steps We first write what we want to find as word phrases.

Then, we represent each phrase in terms of a variable. How much is invested at each rate? Step 3 Next, we make a table showing the amount of money invested, the rates of interest, and the amounts of interest. Step 4 Now, we can write an equation relating the interest from each in- vestment and the total interest received.

The basic idea of solving mixture problems is that the amount or value of the substances being mixed must equal the amount or value of the final mixture. Steps We first write what we want to find as a word phrase. Then, we represent the phrase in terms of a variable. Kilograms of 80c candy: x. Step 3 Next, we make a table showing the types of candy, the amount of each, and the total values of each. Step 3 Next, we make a table or drawing showing the percent of each solu- tion, the amount of each solution, and the amount of pure acid in each solution.

Step 4 We can now write an equation relating the amounts of pure acid before and after combining the solutions. The distributive law can be used to multiply binomials; the FOIL method suggests the four products involved. Solve equations and inequalities Simplify expressions Factor polynomials Graph equations and inequalities Advanced solvers All solvers Tutorials. Partial Fractions. Welcome to Quickmath Solvers! New Example. Help Tutorial.

For example, In either case the result is the same. Our first example involves the product of a monomial and binomial. Example 1 Write 2x x - 3 without parentheses. Solution Applying the distributive property yields When simplifying expressions involving parentheses, we first remove the parentheses and then combine like terms. We begin by removing parentheses to obtain Now, combining like terms yields a - 3a 2. Example 1 a. Using Example 1 , we get If the common monomial is hard to find, we can write each term in prime factored form and note the common factors.

Solution We can write We now see that 2x is a common monomial factor to all three terms. Example 3 Factor out the common monomial, including We will now apply the above procedure for an expression containing variables.

The FOIL method can also be used to square binomials. Consider the following product. We see that the only pair of factors whose product is 12 and whose sum is 7 is 3 and 4. We see that the only pair of factors whose product is 6 and whose sum is -5 is -3 and We see that the only pair of factors whose product is and whose sum is -1 is -4 and 3.

Example 1 Write as a polynomial. Example 2 a. Example 3 a. Solution Above, we determined that this polynomial is factorable. Now, we consider the factorization of a trinomial in which the constant term is negative. It is easiest to factor a trinomial written in descending powers of the variable. 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.

Such a solution is often called a solution using radicals. In the 18 th and 19 th centuries, the great mathematicians, Euler, Lagrange, Eisenstein and Gauss further e x tended our understanding of polynomials and polynomial equations. This led to the development of what is nowdays called modern algebra which is concerned with the study of algebraic structures. In particular, suppose p x is a polynomial with degree greater than 0, and real coefficients,.

The fundamental theorem of algebra is used to show the first of these statements. To obtain the second, we need to know the fact that when we have a polynomial with real coefficients, any comple x roots will occur in pairs, known as conjugate pairs.

This fact can be used to prove the second statement. This requires a little knowledge of comple x numbers. Eisenstein c. Suppose that we can find a prime number p that does not divide the leading coefficient an, but which does divide all of the other coefficients. That is, p x is irreducible. E x plain how to construct a polynomial of arbitrarily large degree that cannot be factored over the rationals. These are called power series. Thus, for e x ample,. The notation n!

Thus 5! Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator. A polynomial with rational coefficients can sometimes be written as a product of lower-degree polynomials that also have rational coefficients. In such cases, the polynomial is said to "factor over the rationals.

Polynomials with rational coefficients always have as many roots, in the complex plane, as their degree; however, these roots are often not rational numbers. In such cases, the polynomial will not factor into linear polynomials. Rational functions are quotients of polynomials.