3.3 Dividing Polynomials

Long Division of Polynomials Dividing polynomials is much like the familiar process of dividing numbers. When we divide 38 by 7, the quotient is 5 and the remainder is 3. We write

Long Division of Polynomials To divide polynomials, we use long division, as follows.

Example 1 – Long Division of Polynomials Divide 6x2 – 26x + 12 by x – 4. Express the result in each of the two forms of division algorithm.

Synthetic Division Synthetic division is a quick method of dividing polynomials; it can be used when the divisor is of the form x – c. In synthetic division we write only the essential parts of the long division. Compare the following long and synthetic divisions, in which we divide 2x3 – 7x2 + 5 by x – 3.

Synthetic Division Long Division Synthetic Division Note that in synthetic division we abbreviate 2x3 – 7x2 + 5 by writing only the coefficients: 2 –7 0 5, and instead of x – 3, we simply write 3.

Synthetic Division (Writing 3 instead of –3 allows us to add instead of subtract, but this changes the sign of all the numbers that appear in the gold boxes.) The next example shows how synthetic division is performed.

Example 2 – Synthetic Division Use synthetic division to divide 2x3 – 7x2 + 5 by x – 3.

The Remainder and Factor Theorems The next theorem shows how synthetic division can be used to evaluate polynomials easily.

Example 3 – Using the Remainder Theorem to Find the Value of a Polynomial Let P(x) = 3x5 + 5x4 – 4x3 + 7x + 3. (a) Find the quotient and remainder when P (x) is divided by x + 2. (b) Use the Remainder Theorem to find P(–2).

The Remainder and Factor Theorems The next theorem says that zeros of polynomials correspond to factors.

Example 4 – Factoring a Polynomial Using the Factor Theorem Let P (x) = x3 – 7x + 6. Show that P (1) = 0, and use this fact to factor P (x) completely.

Example 5 – Finding a Polynomial Given the Graph Find a polynomial of degree 4 that has the given graph.

Real Zeros of Polynomials 3.4 Real Zeros of Polynomials

Real Zeros Of Polynomials The Factor Theorem tells us that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we study some algebraic methods that help us to find the real zeros of a polynomial and thereby factor the polynomial. We begin with the rational zeros of a polynomial.

Rational Zeros of Polynomials To help us understand the next theorem, let’s consider the polynomial P (x) = (x – 2)(x – 3)(x + 4) = x3 – x2 – 14x + 24 From the factored form we see that the zeros of P are 2, 3, and –4. When the polynomial is expanded, the constant 24 is obtained by multiplying (–2)  (–3)  4. This means that the zeros of the polynomial are all factors of the constant term. Factored form Expanded form

Rational Zeros of Polynomials The following generalizes this observation. We see from the Rational Zeros Theorem that if the leading coefficient is 1 or –1, then the rational zeros must be factors of the constant term.

Example 1 – Using the Rational Zeros Theorem Find the rational zeros of P (x) = x3 – 3x + 2.

Rational Zeros of Polynomials The following box explains how we use the Rational Zeros Theorem with synthetic division to factor a polynomial.

Example 2 – Finding Rational Zeros Write the polynomial P(x) = 2x3 + x2 – 13x + 6 in factored form, and find all its zeros.

Descartes’ Rule of Signs To describe this rule, we need the concept of variation in sign. If P (x) is a polynomial with real coefficients, written with descending powers of x (and omitting powers with coefficient 0), then a variation in sign occurs whenever adjacent coefficients have opposite signs. For example, has three variations in sign.

Descartes’ Rule of Signs

Example 3 – Using Descartes’ Rule Use Descartes’ Rule of Signs to determine the possible number of positive and negative real zeros of the polynomial P (x) = 3x6 + 4x5 + 3x3 – x – 3

Upper and Lower Bounds Theorem We say that a is a lower bound and b is an upper bound for the zeros of a polynomial if every real zero c of the polynomial satisfies a  c  b. The next theorem helps us to find such bounds for the zeros of a polynomial.

Example 4 – Upper and Lower Bounds for Zeros of a Polynomial Show that all the real zeros of the polynomial P (x) = x4 – 3x2 + 2x – 5 lie between –3 and 2.

Using Algebra and Graphing Devices to Solve Polynomial Equations

Example 5 – Solving a Fourth-Degree Equation Graphically Find all real solutions of the following equation, rounded to the nearest tenth: 3x4 + 4x3 – 7x2 – 2x – 3 = 0

Complex Zeros and The Fundamental Theorem of Algebra 3.5

The Fundamental Theorem of Algebra and Complete Factorization The following theorem is the basis for much of our work in factoring polynomials and solving polynomial equations. Because any real number is also a complex number, the theorem applies to polynomials with real coefficients as well.

The Fundamental Theorem of Algebra and Complete Factorization The Fundamental Theorem of Algebra and the Factor Theorem together show that a polynomial can be factored completely into linear factors, as we now prove.

Example 1 – Factoring a Polynomial Completely Let P (x) = x3 – 3x2 + x – 3. (a) Find all the zeros of P. (b) Find the complete factorization of P.

Zeros and Their Multiplicities In the Complete Factorization Theorem the numbers c1, c2, . . . , cn are the zeros of P. These zeros need not all be different. If the factor x – c appears k times in the complete factorization of P (x), then we say that c is a zero of multiplicity k. For example, the polynomial P (x) = (x – 1)3(x + 2)2(x + 3)5 has the following zeros: 1 (multiplicity 3), –2 (multiplicity 2), –3 (multiplicity 5)

Zeros and Their Multiplicities The polynomial P has the same number of zeros as its degree: It has degree 10 and has 10 zeros, provided that we count multiplicities. This is true for all polynomials, as we prove in the following theorem.

Zeros and Their Multiplicities The following table gives further examples of polynomials with their complete factorizations and zeros.

Example 2 – Finding Polynomials with Specified Zeros (a) Find a polynomial P (x) of degree 4, with zeros i, –i, 2, and –2, and with P (3) = 25. (b) Find a polynomial Q(x) of degree 4, with zeros –2 and 0, where –2 is a zero of multiplicity 3.

Complex Zeros Come in Conjugate Pairs As you might have noticed from the examples so far, the complex zeros of polynomials with real coefficients come in pairs. Whenever a + bi is a zero, its complex conjugate a – bi is also a zero.

Example 3 – A Polynomial with a Specified Complex Zero Find a polynomial P (x) of degree 3 that has integer coefficients and zeros and 3 – i.

Linear and Quadratic Factors We have seen that a polynomial factors completely into linear factors if we use complex numbers. If we don’t use complex numbers, then a polynomial with real coefficients can always be factored into linear and quadratic factors. We use this property when we study partial fractions. A quadratic polynomial with no real zeros is called irreducible over the real numbers. Such a polynomial cannot be factored without using complex numbers.

Linear and Quadratic Factors

Example 4 – Factoring a Polynomial into Linear and Quadratic Factors Let P (x) = x4 + 2x2 – 8. (a) Factor P into linear and irreducible quadratic factors with real coefficients. (b) Factor P completely into linear factors with complex coefficients.