Finding Equations of Polynomial Functions with Given Zeros
Polynomials are functions of general form 𝑃(𝑥) = 𝑎𝑛𝑥𝑛+ 𝑎𝑛−1 𝑥𝑛−1 + ⋯ + 𝑎2𝑥2+ 𝑎1𝑥 + 𝑎0
(𝑛 ∈ 𝑤ℎ𝑜𝑙𝑒 #′𝑠)
Polynomials can also be written in factored form 𝑃(𝑥)= 𝑎(𝑥 − 𝑧1)(𝑥 − 𝑧2)…(𝑥 − 𝑧𝑖) (𝑎 ∈ ℝ)
Given a list of “zeros”, it is possible to find a polynomial function that has these specific zeros. In fact, there
are multiple polynomials that will work. In order to determine an exact polynomial, the “zeros” and a point
on the polynomial must be provided.
Examples: Practice finding polynomial equations in general form with the given zeros.
Find an* equation of a polynomial with the
following two zeros: 𝑥 = −2, 𝑥 = 4
Step 1: Start with the factored form of a
polynomial.
𝑃(𝑥)= 𝑎(𝑥 − 𝑧1)(𝑥 − 𝑧2)
Step 2: Insert the given zeros and simplify.
𝑃(𝑥)= 𝑎(𝑥 − (−2))(𝑥 − 4)
𝑃(𝑥)= 𝑎(𝑥 + 2)(𝑥 − 4)
Step 3: Multiply the factored terms
together.
𝑃(𝑥)= 𝑎(𝑥2− 2𝑥 − 8)
Step 4: The answer can be left with the
generic “𝑎”, or a value for “𝑎”can be chosen,
inserted, and distributed.
i.e. if 𝑎 = 1, 𝑡ℎ𝑒𝑛 𝑃(𝑥) = 𝑥2− 2𝑥 − 8
i.e. if 𝑎 = −2, 𝑡ℎ𝑒𝑛 𝑃(𝑥)= −2𝑥2+ 4𝑥 + 16
*Each different choice for “𝑎” will result in a
distinct polynomial. Thus, there are an
infinite number of polynomials with the two
zeros 𝑥 = −2 𝑎𝑛𝑑 𝑥 = 4.
Find the equation of a polynomial with the
following zeroes: 𝑥 = 0, −√2, √2 that goes
through the point (−2, 1).
Step 1: Start with the factored form of a
polynomial.
𝑃(𝑥)= 𝑎(𝑥 − 𝑧1)(𝑥 − 𝑧2)(𝑥 − 𝑧3)
Step 2: Insert the given zeros and simplify.
𝑃(𝑥)= 𝑎(𝑥 − 0)(𝑥 − (−√2))(𝑥 − √2)
𝑃(𝑥)=𝑎𝑥(𝑥 + √2)(𝑥 − √2)
Step 3: Multiply the factored terms together
𝑃(𝑥)= 𝑎(𝑥3− 2𝑥)
Step 4: Insert the given point (−2, 1) to
solve for “𝑎 “.
1 = 𝑎[(−2)3− 2(−2)]
1 = 𝑎[−8 + 4]
1 = −4𝑎
𝑎 = −1
4
Step 5: Insert the value for “𝑎” into the
polynomial, distribute, and re-write the
function.
𝑃(𝑥)= − 1
4(𝑥3− 2𝑥)= − 1
4𝑥3+1
2𝑥
Denote the given zeros as 𝑧1 𝑎𝑛𝑑 𝑧2
Denote the given zeros as 𝑧1, 𝑧2𝑎𝑛𝑑 𝑧3
