02.6 F Rife

Descartes's Rule of Signs

1. The number of positive zeros of F is equal to the number of variations of sign of F(x) or is less than that number by an even integer. 2. The number of negative zeros of F is equal to the number of variations of sign of F(-x) or is less than that number by an even integer. 3. When using Descartes's Rule, a zero of multiplicity m should be counted as m zeros.

Zeros of a Polynomial

2.6

Example 3: Find the polynomial P(x) with Degree 4; leading coefficient 2, zeros: 5-i, 3i.

According to the conjugate pairs theorem, if 5-1 is a zero of the function, then 5+i is also a zero of the function. Likewise, if 3i is a zero, then -3i must also be a zero. Writing out the factors, we get 2(x-3i)(x+3i)(x-(5-i))(x-(5+i)). Keeping in mind the fact that i^2=-1, distribute and combine all of the factors, until you end up with one final equation. P(x)=2x^4-20x^3+70x^2-180x+468.

Odd-degree polynomials with real zeros

Any polynomial P(x) of odd degree with real coefficients must have at least one real zero.

Number of zeros theorem

Any polynomial of degree n has exactly n zeros, provided a zero of multiplicity k is counted k times.

Example 4: A polynomial of degree 7 has the following zeros 3 of multiplicity 1, 5 of multiplicity 2, -i of multiplicity 1, and 4+i of multiplicity 1. Determine the number of zeros, then write the zeros of P(x).

Because of the conjugate pairs theorem, we know that if -i is a zero then i must also be a zero. Likewise, if 4+i is a zero then 4-i must also be a zero. Hence, the zeros of P(x) are... 3, 5, 5, -i, i, 4+i, 4-i.

Example 2: Determine the upper and lower bounds on the zeros of the function f(x)=3x^3-x^2+9x-3.

First we must find all of the possible zeros of the polynomial function. The factors of -3 are +/- 1 and +/- 3. The factors of 3 are also +/- 1 and +/- 3. So the possible zeros are +/- 1, +/- 1/3, +/- 3. When using synthetic division to find the upper bound start with the smallest possible positive value and work your way up. To find the upper bound, all values in the last row must be positive or zero. The upper bound then becomes 1/3. To find the lower bound start with the smallest possible negative value and repeat the synthetic division process. All values in the last row must alternate in sign, where 0 can be positive or negative. The lower bound then becomes -1/3.

Upper bound

If F(x) has no greater zeros than a number k, then k is called an upper bound on the zeros of F(x).

Lower Bound

If F(x) has no zeros less than a number k, then k is the called a lower bound on the zeros of F(x).

Factorization Theorem for Polynomials

If P(x) is a complex polynomial of degree n>=1, it can be factored into n (not necessarily distinct) linear factors of the form P(x)=a(x-r1)(x-r2)(x-r3)... where a, r1, r2, r3... are complex numbers.

Example 1: Determine the number of positive and negative zeros of the function f(x)=5x^3-2x^2-3x+4.

In f(x) there are two variations in sign (5x^3-2x^2, and -3x+4) so the number of positive zeros must be either 2 or 2-2=0. f(-x)=-5x^3-2x^2+3x+4. Since there is only one variation in sign (-2x^2+3x) the number of negative zeros can only be 1 since an even integer cannot be subtracted from 1 and still be greater than or equal to zero.

Variation of Sign

Occurs when the sign of two consecutive terms differ.

Example 6: Given that 3i is a zero of the function P(x)=x^4+x^3+9x^2+9x, find all other zeros.

Since 3i is a zero of the function, -3i must also be a zero. The first step is to multiply these two factors together. (x-3i)(x+3i)= x^2-9i^2. i^2=-1, so x^2-9i^2= x^2+9 Then we use long division to divide P(x) by x^2+9 to get... x^2+x. Our factors now become (x^2+x)(x^2+9). First factoring and solving x^2+x=0, we get the factors x(x+1), which solves to be x=0 and x=-1. Then factoring x^2+9, we get the factors x= +/- square root of -9, which are 3i and -3i. So, the zeros of P(x) are 0, -1, 3i, -3i.

Example 5: Find all solutions of the equation in the complex number system. (x-2)(x-3i)(x+3i)=0.

Since the equation is already in factored form, you simply have to solve each factor for 0. x-2=0 x=2 x-3i=0 x=3i x+3i=0 x=-3i the solution set is {-3i, 3i, and 2}

Steps For Gathering Information and locating the real zeros of a polynomial function

Step 1: Find the maximum number of real zeros by using the degree of the polynomial function. Step 2: Find the possible number of positive and negative zeros by using the Descartes's Rule of Signs. Step 3: Write the set of possible rational zeros Step 4: Test the smallest positive integer in the set in step 3, then the next larger, and so on, until an integer zero or an upper bound of the zeros is found. If a zero is found, use the function represented by the depressed equation in further calculations. If an upper bound is found, discard all larger numbers in the set of possible rational zeros.

Steps For Gathering Information and locating the real zeros of a polynomial function (cont.)

Step 5: Test the positive fractions that remain in the set in step 3 after considering any bound that has been found. Step 6: Use modified steps 4 and 5 for negative numbers in the set in step 3. Step 7: If a depressed equation is quadratic, use any method (including the quadratic formula) to solve this equation.

Rules for Bounds

Suppose F(x) is synthetically divided by x-k. 1. If k>0 and each number in the last row is zero or positive, then k is an upper bound on the zeros of F(x). 2. If k<0 and numbers in the last row alternate in sign (zeros in the last row can be regarded as positive or negative), then k is a lower bound on the zeros of F(x).

Rational Zeros (Roots) Theorem

To find the total number of possible rational zeros, divide p/q, where p is the factors of the constant term and q is the factors of the leading coefficient.

02.6 F RIfe

Zeros of a Polynomial

Fundamental Theorem of Algebra

if f(x) is a polynomial of degree n, where n>0, then f has at least one zero in the complex number system

Conjugate Pairs Theorem

in a polynomial function, if (a+bι) is a zero, then (a-bι) is a zero