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Grade 11 · unit review

Review: Algebra2: Polynomial Division and Zeros

Polynomial functions connect symbolic factors with graph features. Division, remainder evaluation, real zeros and multiplicity reveal structure without requiring every coefficient to be expanded, but domains and repeated roots still need attention.

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Unit review questions
1. Find the remainder when P(x)=x²+2x+3 is divided by x−7.
Need a hint?
  • Dividing a polynomial by x−a leaves a constant remainder equal to its value at a.
  • Identify the divisor’s zero before evaluating.
2. Divide x²−49 by x−7. Give the polynomial quotient.
Need a hint?
  • Polynomial division asks which factor times the divisor reconstructs the original polynomial, possibly with a remainder.
  • A polynomial quotient can be defined everywhere while the original fraction is not.
3. P(x)=(x−8)(x−2)(x+2). List all real zeros.
Need a hint?
  • A cubic already written as a product of linear factors exposes its real zeros through the zero-product property.
  • Match each factor’s sign correctly.
4. For y=(x−7)^3(x+8), what does the graph do at x=7?
Need a hint?
  • A repeated even-power factor stays the same sign on both sides of its zero.
  • This is local behavior at the named root.

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Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Find the remainder when P(x)=x²+2x+3 is divided by x−9. Candidate answers: (A) 101 (B) 102 (C) 103 (D) 204. 2. Divide x²−81 by x−9. Give the polynomial quotient. Candidate answers: (A) x+81 (B) x+9 (C) x²+9 (D) x−9. 3. P(x)=(x−10)(x−2)(x+2). List all real zeros. Candidate answers: (A) 0, 2, 10 (B) 10 only (C) −10, −2, 2 (D) −2, 2, 10. 4. For y=(x−9)^3(x+10), what does the graph do at x=9? Candidate answers: (A) Cannot have value zero there (B) Crosses the x-axis there (C) Has a vertical asymptote there (D) Touches the x-axis and turns without crossing.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 102. The remainder theorem gives P(9)=9²+2(9)+3=102. The divisor x−9 sets the substitution value to positive 9. 2. x+9. Factor the numerator as (x−9)(x+9). Polynomial division gives quotient x+9 and remainder 0; the rational expression itself is undefined at x=9. 3. −2, 2, 10. Each factor can be zero: x=10, x=2, or x=−2. These are distinct because 10>2. Multiplying the factors would not change these zeros. 4. Crosses the x-axis there. The zero has multiplicity 3. An even factor keeps its sign across the root and touches; an odd factor changes sign and crosses. The other factor stays positive nearby, so this graph crosses.

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