Zeros of Polynomials

Polynomial Type of Coefficient

5x3 + 3x2 + (2 + 4i) + i complex

5x3 + 3x2 + √2x – π real

5x3 + 3x2 + ½ x – ⅜ rational

5x3 + 3x2 + 8x – 11 integer

Zeros of a Polynomial

Zeros (Solutions)

Real Zeros Complex Zeros

Rational or Irrational ZerosComplex Number and its Conjugate

Rational Zero Theorem If the polynomial

f(x) = anxn + an-1xn-1 + . . . + a1x + a0

has integer coefficients, then every rational

zero of f(x) is of the form p

qwhere p is a factor of the constant a0

and q is a factor of the leading coefficient an.

Rational Root (Zero) Theorem

• If “q” is the leading coefficient and “p” is the constant term of a polynomial, then the only possible rational roots are + factors of “p” divided by + factors of “q”. (p / q)

• Example: • To find the POSSIBLE rational roots of f(x), we

need the FACTORS of the leading coefficient (6 for this example) and the factors of the constant term (4, for this example). Possible rational roots are

41246)( 35 xxxxf

factors of 1, 2, 4 1 1 1 2 41,2,4, , , , ,

factors of 1, 2, 3, 6 2 3 6 3 3

p

q

• List all possible rational zeros of

f(x) = x3 + 2x2 – 5x – 6.

• List all possible rational zeros of

f(x) = 4x5 + 12x4 – x – 3.

How do we know which possibilities are really zeros (solutions)?

• Use trial and error and synthetic division to see if one of the possible zeros is actually a zero.

• Remember: When dividing by x – c, if the remainder is 0 when using synthetic division, then c is a zero of the polynomial.

• If c is a zero, then solve the polynomial resulting from the synthetic division to find the other zeros.

• Find all zeros of f(x) = x3 + 8x2 + 11x – 20.

Finding the Rational Zeros of a Polynomial1. List all possible rational zeros of the

polynomial using the Rational Zero Theorem.

2. Use synthetic division on each possible rational zero and the polynomial until one gives a remainder of zero. This means you have found a zero, as well as a factor.

3. Write the polynomial as the product of this factor and the quotient.

4. Repeat procedure on the quotient until the quotient is quadratic.

5. Once the quotient is quadratic, factor or use the quadratic formula to find the remaining real and imaginary zeros.

• Find all zeros of f(x) = x3 + x2 - 5x – 2.

How many zeros does a polynomial with rational coefficients have?

• An nth degree polynomial has a total of n zeros. Some may be rational, irrational or complex.

• Because all coefficients are RATIONAL, irrational roots exist in pairs (both the irrational # and its conjugate). Complex roots also exist in pairs (both the complex # and its conjugate).

• If a + bi is a root, a – bi is a root• If is a root, is a root.• NOTE: Sometimes it is helpful to graph the

function and find the x-intercepts (zeros) to narrow down all the possible zeros.

ba ba

• Solve: x4 – 6x3 + 22x2 - 30x + 13 = 0.

Remember…• Complex zeros come in pairs as

complex conjugates: a + bi, a – bi

• Irrational zeros come in pairs.

a c b , a c b

Practice

Find a polynomial function (in factored form) of degree 3 with 2 and i as zeros.

More Practice – woohoo!Find a polynomial function (in factored

form) of degree 5 with -1/2 as a zero with

multiplicity 2, 0 as a zero of multiplicity 1,

and 1 as a zero of multiplicity 2.

Double woohoo!• Find a third-degree polynomial function

f(x) with real coefficients that has -3 and i as zeros and such that f(1) = 8.

Extra Fun!

• Suppose that a polynomial function of degree 4 with rational coefficients has i and

–3 +√3 as zeros. Find the other zero(s).