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Math Analysis Notes
Section 3.3
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Section 3.3: Real Zeros of Polynomials
Big Idea: This section has 3 theorems that help you find exact values for real zeros of polynomials with
integer coefficients.
Big Skill: You should be able to find the zeros of a polynomial using these techniques.
p
, where p is a factor
q
of the constant coefficient a0, and q is a factor of the leading coefficient an. Note: this theorem does not
include irrational or complex zeros; only rational zeros.
2. Brute force method for finding rational zeros of a polynomial:
a. List all possible rational zeros of the polynomial using the Rational Zeros Theorem.
b. Test each possible zero using synthetic division. A remainder of 0 means you have found a zero,
and the bonus is that the other numbers you obtained from the synthetic division are the
coefficients of the next polynomial you need to factor.
c. Repeat until you find a quotient that is quadratic or otherwise factors easily.
3. Descartes’ Rule of Signs: Let P be a polynomial with real coefficients.
a. The number of positive real zeros of P(x) is either equal to the number of variations in sign in
P(x) or is less than that by a positive even whole number.
b. The number of negative real zeros of P(x) is either equal to the number of variations in sign in
P(-x) or is less than that by a positive even whole number.
c. This Rule can be used to help limit the number of possible zeros to check…
4. Upper and Lower Bounds Theorem: Let P be a polynomial with real coefficients.
a. If we divide P(x) by x – b (with b > 0) using synthetic division, and if the row that contains the
quotient and remainder has no negative entry, then b is an upper bound for the real zeros of P.
b. If we divide P(x) by x – a (with a < 0) using synthetic division, and if the row that contains the
quotient and remainder has entries that are alternatively nonpositive and nonnegative, then a is a
lower bound for the real zeros of P.
1. Rational Zeros Theorem: All rational zeros of a polynomial P are of the form
Steps for factoring a polynomial:
1. Make a list of all possible rational zeros using the rational zeros theorem.
2. Order the list and remove any redundant values.
3. Apply Descartes’ Rule of signs to predict how many positive and negative zeros you can expect; if you
find that no roots of a given sign are predicted, then you just cut your work in half.
4. Perform synthetic division on the polynomial, starting with the smallest roots.
a. If the remainder is zero, then you found a root; check to see if it is a multiple root, and re-write
the polynomial with that root factored out. Only perform synthetic division on the lower-order
quotient.
b. If the remainder is not zero, look at the signs of the quotient; the upper and lower bounds
theorem tells you when you can stop looking based on the signs of the numbers in the quotient
row.
5. Use the quadratic formula when you get down to a quadratic quotient.
Math Analysis Notes
Section 3.3
6. Practice:
Factor P(x) = 2x3 + 7x2 + 4x – 4.
Factor P(x) = 4x5 – 18x4 – 6x3 + 91x2 – 60x + 9.
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