7.1: Finding Rational Solutions of Polynomial Equations

Rational Root Theorem Used to find all possible rational roots (i.e., zeros). Steps: Find the factors of the constant term (i.e., p) Find the factors of the coefficient of the leading term (i.e., q) Write all factors as ± Find all possibilities of 𝑝 𝑞 𝒑 𝒒 = ±1,±2,±5,±10 ±1,±3,±7,±21 =±𝟏,±𝟐,±𝟓,±𝟏𝟎,± 𝟏 𝟑 ,± 𝟏 𝟕 ,± 𝟏 𝟐𝟏 ,± 𝟐 𝟑 ,± 𝟐 𝟕 ,± 𝟐 𝟐𝟏 ,± 𝟓 𝟑 ,± 𝟓 𝟕 ,± 𝟓 𝟐𝟏 ,± 𝟏𝟎 𝟑 ,± 𝟏𝟎 𝟕 ,± 𝟏𝟎 𝟐𝟏

Ex #1 Use the Rational Root Theorem to list all possible rational roots for each equation. a) 2 𝑥 3 − 𝑥 2 +2𝑥+5=0 b) 3 𝑥 3 +7 𝑥 2 +6𝑥−8=0 𝒑 𝒒 = ±1,±5 ±1,±2 =±𝟏,± 𝟏 𝟐 ,±𝟓,± 𝟓 𝟐 𝒑 𝒒 = ±1,±2,±4,±8 ±1,±3 =±𝟏,± 𝟏 𝟑 ,±𝟐,± 𝟐 𝟑 ,±𝟒,± 𝟒 𝟑 ,±𝟖,± 𝟖 𝟑

Finding Actual Rational Roots Steps: List all possible rational roots (i.e., 𝑝 𝑞 ) Test each one using synthetic division When you find one that gives you a remainder of 0, you know that it’s a zero. The rest of the entries in the synthetic division tell you the quotient, or remaining factor. Finish factoring what remains to find other roots.

Ex #2 Use the Rational Root Theorem to list all possible rational roots for the equation. Then find any actual rational roots. (a) 𝑥 3 −4 𝑥 2 −7𝑥+10=0

Cont. (b) 2 𝑥 3 −5 𝑥 2 −28𝑥+15=0

(Cont.) On Your Own (c) 3 𝑥 3 −12 𝑥 2 +3𝑥+18=0 Hint: Factor out GCF first! Answer: Possible rational roots = ±1,±2,±3,±6 ±1 =±1,±2,±3,±6 Actual rational roots = 2, −1, 3

Engineering Application Ex: A pen company is designing a gift container for their new premium pen. The marketing department has designed a pyramidal box with a rectangular base. The base width is 1 inch shorter than its base length and the height is 3 inches taller than 3 times the base length. The volume of the box must be 6 cubic inches. What are the dimensions of the box?

Solution:

On Your Own: A box company is designing a new rectangular gift container. The marketing department has designed a box with a width 2 inches shorter than its length and a height 3 inches taller than its length. The volume of the box must be 56 cubic inches. What are the dimensions of the box?

Conjugate Root Theorem If we know a complex number is a root, then its complex conjugate is also a root! Ex: If 2−4𝑖 is a root, so is 2+4𝑖 Ex: If 2𝑖 is a root, so is −2𝑖

Ex #3 Write a polynomial function with rational coefficients so that it has the roots: (a) −3 and 2𝑖

(Cont.) On Your Own (b) 1 and −𝑖