The Remainder Theorem - Evaluate Functions and Divide Polynomials

Oct 8, 2026

Worksheets for Infinite Algebra 2 on the Remainder Theorem. Students evaluate multiple polynomial functions at given values and determine whether specified binomials are factors of the corresponding polynomials, recording Yes/No responses. The sheet also includes polynomial division problems, requiring simplified quotient and remainder forms (R ...). Designed for practice consolidating theorem-based evaluation, factor checks, and long-division style outcomes across varied variables and degrees.

LanguageEnglish

CategoryExam

FormatPDF

Pages4

Reading Time10 min

AccessFree to download

Evaluate each function at the given value

State if the given binomial is a factor of the given polynomial

Divide

Kuta Software-Infinite Algebra 2 Name___________________________________ The Remainder Theorem Date________________ Period____ Evaluate each function at the given value. 1) f (x) = −x3 + 6x − 7 at x = 2 2) f (x) = x 3 + x 2 − 5x − 6 at x = 2 3) f (a) = a 3 + 3a2 + 2a + 8 at a = −3 4) f (a) = a 3 + 5a2 + 10a + 12 at a = −2 5) f (a) = a4 + 3a3 − 17a2 + 2a − 7 at a = 3 6) f (x) = x 5 − 47x3 − 16x2 + 8x + 52 at x = 7 State if the given binomial is a factor of the given polynomial. 7) (k3 − k2 − k − 2) ÷ (k − 2) 8) (b4 − 8b3 − b2 + 62b − 34) ÷ (b − 7) 9) (n4 + 9n3 + 14n2 + 50n + 9) ÷ (n + 8) 10) ( p4 + 6 p3 + 11 p2 + 29 p − 13) ÷ ( p + 5) 11) ( p4 − 8 p3 + 10 p2 + 2 p + 4) ÷ ( p − 2) 12) (n5 − 25n3 − 7n2 − 37n − 18) ÷ (n + 5) 13) (x5 + 6x4 − 3x2 − 22x − 29) ÷ (x + 6) 14) (n4 + 10n3 + 21n2 + 6n − 8) ÷ (n + 2) Divide. 15) ( p4 + 5 p3 − 11 p2 − 25 p + 29) ÷ ( p + 6) 16) (8k3 − 66k2 + 14k + 8) ÷ (k − 8) 17) (x4 + 11x3 + 33x2 + 24x + 32) ÷ (x + 6) 18) (6v3 + 42v2 − 50v − 20) ÷ (v + 8) 19) (6b4 + 53b3 + 32b2 − 61b + 19) ÷ (b + 8) 20) (4n3 − 9n2 + 9n + 3) ÷ (n − 1) 21) (6a3 + 20a2 − 15a + 9) ÷ (a + 4) 22) (n4 − 6n3 − 10n2 + 20n + 15) ÷ (n + 2) 23) ( p5 + 8 p4 + 2 p2 + 19 p + 16) ÷ ( p + 8) 24) (x4 − 2x3 − 16x2 + 28x + 9) ÷ (x − 4) 25) (r5 + 6r4 − 13r3 − 5r2 − 8r + 14) ÷ (r − 2) 26) (8v5 + 32v4 + 5v + 20) ÷ (v + 4) Kuta Software-Infinite Algebra 2 The Remainder Theorem Evaluate each function at the given value. 1) f (x) = −x3 + 6x − 7 at x = 2 −3 Name___________________________________ Date________________ Period____ 2) f (x) = x 3 + x 2 − 5x − 6 at x = 2 −4 3) f (a) = a 3 + 3a2 + 2a + 8 at a = −3 4) f (a) = a 3 + 5a2 + 10a + 12 at a = −2 2 4 5) f (a) = a4 + 3a3 − 17a2 + 2a − 7 at a = 3 6) f (x) = x 5 − 47x3 − 16x2 + 8x + 52 at x = 7 8 10 State if the given binomial is a factor of the given polynomial. 7) (k3 − k2 − k − 2) ÷ (k − 2) 8) (b4 − 8b3 − b2 + 62b − 34) ÷ (b − 7) Yes No 9) (n4 + 9n3 + 14n2 + 50n + 9) ÷ (n + 8) 10) ( p4 + 6 p3 + 11 p2 + 29 p − 13) ÷ ( p + 5) No No 11) ( p4 − 8 p3 + 10 p2 + 2 p + 4) ÷ ( p − 2) 12) (n5 − 25n3 − 7n2 − 37n − 18) ÷ (n + 5) Yes No 13) (x5 + 6x4 − 3x2 − 22x − 29) ÷ (x + 6) 14) (n4 + 10n3 + 21n2 + 6n − 8) ÷ (n + 2) No Yes Divide. 15) ( p4 + 5 p3 − 11 p2 − 25 p + 29) ÷ ( p + 6) 16) (8k3 − 66k2 + 14k + 8) ÷ (k − 8) p 3 − p2 − 5 p + 5, R −1 8k2 − 2k − 2, R −8 17) (x4 + 11x3 + 33x2 + 24x + 32) ÷ (x + 6) 18) (6v3 + 42v2 − 50v − 20) ÷ (v + 8) x 3 + 5x2 + 3x + 6, R −4 6v2 − 6v − 2, R −4 19) (6b4 + 53b3 + 32b2 − 61b + 19) ÷ (b + 8) 20) (4n3 − 9n2 + 9n + 3) ÷ (n − 1) 6b3 + 5b2 − 8b + 3, R −5 4n2 − 5n + 4, R 7 21) (6a3 + 20a2 − 15a + 9) ÷ (a + 4) 22) (n4 − 6n3 − 10n2 + 20n + 15) ÷ (n + 2) 6a2 − 4a + 1, R 5 n 3 − 8n2 + 6n + 8, R −1 23) ( p5 + 8 p4 + 2 p2 + 19 p + 16) ÷ ( p + 8) 24) (x4 − 2x3 − 16x2 + 28x + 9) ÷ (x − 4) p4 + 2 p + 3, R −8 x 3 + 2x2 − 8x − 4, R −7 25) (r5 + 6r4 − 13r3 − 5r2 − 8r + 14) ÷ (r − 2) 26) (8v5 + 32v4 + 5v + 20) ÷ (v + 4) r4 + 8r3 + 3r2 + r − 6, R 2 8v4 + 5 Create your own worksheets like this one with Infinite Algebra 2. Free trial available [at KutaSoftware.com](at KutaSoftware.com)

FAQs

How do you use the Remainder Theorem to evaluate a polynomial at a specific value?

Substitute the given value into the polynomial. The result gives the remainder associated with that input under the Remainder Theorem setup.

What does it mean to check whether a binomial is a factor of a polynomial?

A binomial (like (x+a)) is a factor exactly when the corresponding remainder is 0 after evaluating using the theorem.

What output is expected for the “Divide” problems?

Each division result should be expressed as a quotient in simplified form together with a remainder written as R followed by the remainder value.

Oct 8, 2026

Uploaded by Maya Linwood