Analyzing and Solving Polynomial Equations - State the number of complex/real/imaginary roots, then find all roots

Oct 8, 2026

Uploaded by Miles

Worksheets for Infinite Algebra 2 focus on analyzing polynomial equations and determining root characteristics. For each listed equation, the worksheet instructs students to state the number of complex roots and the possible counts of real and imaginary roots, including how many real roots are positive or negative and which rational roots are possible. After classification, students find all roots, including complex solutions with explicit sets of values for each polynomial.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Polynomial Root Analysis and Solving

Root counting: complex, real, imaginary

Positive/negative real roots and possible rational roots

Finding all roots for each equation

Kuta Software-Infinite Algebra 2 Name Analyzing and Solving Polynomial Equations Date Period State the number of complex roots, the possible number of real and imaginary roots, the possible number of positive and negative roots, and the possible rational roots for each equation. Then find all roots. 1) x 4 − 5x2 − 36 = 0 2) x 3 + 3x2 − 14x − 20 = 0 3) x 3 − 2x2 + 3x − 6 = 0 4) x 4 − 14x2 + 45 = 0 5) x 4 + 6x2 + 8 = 0 6) x 4 + 3x2 − 18 = 0 7) x 3 − 1 = 0 8) x 3 + 3x2 − x − 3 = 0 9) x 3 − 2x2 − 3x + 6 = 0 10) x 6 − 2x4 − 4x2 + 8 = 0 11) x 5 + 2x4 + 11x3 + 22x2 + 24x + 48 = 0 12) x 6 + 5x4 − 4x2 − 20 = 0 13) x 6 − x 4 − x 2 + 1 = 0 14) x 8 − 26x4 + 25 = 0 Kuta Software-Infinite Algebra 2 Name Analyzing and Solving Polynomial Equations Date Period State the number of complex roots, the possible number of real and imaginary roots, the possible number of positive and negative roots, and the possible rational roots for each equation. Then find all roots. 1) x 4 − 5x2 − 36 = 0 \# of complex roots: 4 Possible \# of real roots: 4, 2, or 0 Possible \# of imaginary roots: 4, 2, or 0 Possible \# positive real roots: 1 Possible \# negative real roots: 1 Possible rational roots: ± 1, ± 2, ± 3, ± 4, ± 6, ± 9, ± 12, ± 18, ± 36 Roots: {2i, −2i, 3, −3} 3) x 3 − 2x2 + 3x − 6 = 0 \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible \# positive real roots: 3 or 1 Possible \# negative real roots: 0 Possible rational roots: ± 1, ± 2 , ± 3, ± 6 Roots: {2, i 3 , −i 3 } 5) x 4 + 6x2 + 8 = 0 \# of complex roots: 4 Possible \# of real roots: 4, 2, or 0 Possible \# of imaginary roots: 4, 2, or 0 Possible \# positive real roots: 0 Possible \# negative real roots: 0 Possible rational roots: ± 1, ± 2 , ± 4, ± 8 Roots: {2i, −2i, i 2 , −i 2 } 7) x 3 − 1 = 0 \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible \# positive real roots: 1 Possible \# negative real roots: 0 Possible rational roots: ± 1 Roots: { 1, −1~~ ~~+2i~~ ~~3~~ ~~ , −1~~ ~~−2i~~ ~~3~~ ~~ } 2) x 3 + 3x2 − 14x − 20 = 0 \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible \# positive real roots: 1 Possible \# negative real roots: 2 or 0 Possible rational roots: ± 1, ± 2, ± 4, ± 5, ± 10, ± 20 Roots: {−5, 1 + 5 , 1 − 5 } 4) x 4 − 14x2 + 45 = 0 \# of complex roots: 4 Possible \# of real roots: 4, 2, or 0 Possible \# of imaginary roots: 4, 2, or 0 Possible \# positive real roots: 2 or 0 Possible \# negative real roots: 2 or 0 Possible rational roots: ± 1, ± 3, ± 5, ± 9, ± 15, ± 45 Roots: { 5 , − 5 , 3, −3 } 6) x 4 + 3x2 − 18 = 0 \# of complex roots: 4 Possible \# of real roots: 4, 2, or 0 Possible \# of imaginary roots: 4, 2, or 0 Possible \# positive real roots: 1 Possible \# negative real roots: 1 Possible rational roots: ± 1, ± 2 , ± 3, ± 6, ± 9, ± 18 Roots: { 3 , − 3 , i 6 , −i 6 } 8) x 3 + 3x2 − x − 3 = 0 \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible \# positive real roots: 1 Possible \# negative real roots: 2 or 0 Possible rational roots: ± 1, ± 3 Roots: {−3, 1, −1} 9) x 3 − 2x2 − 3x + 6 = 0 \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible \# positive real roots: 2 or 0 Possible \# negative real roots: 1 Possible rational roots: ± 1, ± 2 , ± 3, ± 6 Roots: {2, 3 , − 3 } 11) x 5 + 2x4 + 11x3 + 22x2 + 24x + 48 = 0 \# of complex roots: 5 Possible \# of real roots: 5, 3, or 1 Possible \# of imaginary roots: 4, 2, or 0 Possible \# positive real roots: 0 Possible \# negative real roots: 5, 3, or 1 Possible rational roots: ± 1, ± 2, ± 3, ± 4, ± 6, ± 8, ± 12, ± 16, ± 24, ± 48 Roots: {−2, i 3 , −i 3 , 2i 2 , −2i 2 } 13) x 6 − x 4 − x 2 + 1 = 0 \# of complex roots: 6 Possible \# of r

FAQs

What root information must be determined before solving each polynomial equation?

The worksheet asks for the number of complex roots and the possible numbers of real and imaginary roots, plus the possible counts of positive and negative real roots and the possible rational roots.

How does the worksheet handle rational roots?

For each equation, it lists a set of possible rational roots (including ± values) and then provides the actual roots after solving.

What output does the worksheet require after classifying the roots?

After stating root counts and possible rational roots, it requires finding all roots, including complex roots, shown as a complete roots set for each polynomial.

Oct 8, 2026

Uploaded by Miles