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NY Regents Algebra II Zeros of Polynomial Functions — practice questions

Explore 5 free Zeros of Polynomial Functions questions in NY Regents Algebra II Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Zeros of Polynomial Functions questions and exercises with answers

Showing questions 1–5 of 5.

The Fundamental Theorem of Algebra guarantees a polynomial of degree above 0 has at least what?Read the answer and worked solution →Is every complex zero of a polynomial an imaginary number?Read the answer and worked solution →Divide f(x) = x cubed - 4x squared + 2x + 7 by (x - 3). What remainder does the Remainder Theorem give?Read the answer and worked solution →For f(x) = 3x cubed + 2x squared - 7x + 4, which value is NOT a possible rational zero?Read the answer and worked solution →Apply Descartes' Rule of Signs to f(x) = x to the fourth - 3x cubed + 2x squared + x - 6. How many negative real zeros can it have?Read the answer and worked solution →

Worked practice questions

The Fundamental Theorem of Algebra guarantees a polynomial of degree above 0 has at least what?

  1. One rational zero
  2. Two real zeros
  3. Two turning points
  4. One complex zero
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Correct answer: D. One complex zero

1. Some polynomials have no real zeros at all, so a guarantee about real zeros would be false. 2. The Fundamental Theorem of Algebra promises at least one complex zero for any degree above 0. 3. Applying it repeatedly splits a degree n polynomial into n linear factors. 4. Rational zeros are not guaranteed, since the Rational Zero Theorem only supplies candidates to test. _Source: OpenStax Precalculus (CC BY 4.0), section 3.6 Zeros of Polynomial Functions_

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Is every complex zero of a polynomial an imaginary number?

  1. No, real numbers are complex too
  2. Yes, all complex zeros are imaginary
  3. No, only rational zeros count here
  4. Yes, unless the degree is even
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Correct answer: A. No, real numbers are complex too

1. A complex number has the form a + bi, and b is allowed to be 0. 2. When b is 0 the number is real, so every real number is also a complex number. 3. So a guarantee of a complex zero does not promise an imaginary one. _Source: OpenStax Precalculus (CC BY 4.0), section 3.6 Zeros of Polynomial Functions_

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Divide f(x) = x cubed - 4x squared + 2x + 7 by (x - 3). What remainder does the Remainder Theorem give?

  1. A remainder of 0
  2. A remainder of 7
  3. A remainder of 10
  4. A remainder of 4
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Correct answer: D. A remainder of 4

1. The Remainder Theorem says that dividing by x minus k leaves the remainder f(k). 2. Here the divisor is x - 3, so k is 3. 3. Evaluate f(3), starting with 3 cubed, which is 27. 4. Next comes negative 4 times 9, which is negative 36. 5. Next comes 2 times 3, which is 6, and finally the constant 7. 6. Adding gives 27 minus 36 plus 6 plus 7, which is 4. 7. So the remainder is 4, and it is not 0, so x - 3 is not a factor. 8. A remainder of 0 would assume the divisor is a factor without testing it. 9. A remainder of 7 reads off the constant term instead of evaluating at 3. 10. A remainder of 10 comes from using k equal to negative 3 by mistake. _Source: OpenStax Precalculus (CC BY 4.0), section 3.6 Zeros of Polynomial Functions_

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More Mathematics topics

Complex NumbersExponential and Logarithmic ModelsGraphs of Polynomial FunctionsPower and Polynomial FunctionsRational FunctionsProperties of LogarithmsQuadratic FunctionsDividing Polynomials