Polynomial Functions and Complex Zeros - AP Precalculus

1. What is the complex conjugate of $7 - 5i$?

   $7 + 5i$. Change the sign of the imaginary part.

2. Find the zeros of $x^2 - 9$.

   $x = 3, -3$. Factor as $(x-3)(x+3)$ and set each factor to zero.

3. Determine the degree of $3x^4 + 2x^2 - 7$.

   4. The highest power term determines the degree.

4. What is the factor theorem?

   $x - c$ is a factor if $f(c) = 0$. Direct consequence of the remainder theorem when remainder is zero.

5. Simplify $(1 + 2i)^2$.

   $-3 + 4i$. Expand: $(1+2i)^2 = 1 + 4i + 4i^2 = 1 + 4i - 4$.

6. Find the zero of $x^2 - 4x + 4$.

   $x = 2$. This is $(x-2)^2$, so $x = 2$ is a repeated root.

7. State the Remainder Theorem.

   The remainder of $f(x)$ divided by $x - c$ is $f(c)$. Fundamental theorem for polynomial division and evaluation.

8. Find the product $(3 + 2i)(3 - 2i)$.

   13. Conjugate pairs multiply to give $a^2 + b^2 = 9 + 4 = 13$.

9. State the Conjugate Root Theorem.

   If $a + bi$ is a root, $a - bi$ is also a root. For polynomials with real coefficients, complex roots come in conjugate pairs.

10. State the definition of a monomial.

    A polynomial with one term. Simplest polynomial with exactly one term.

11. What is the multiplicity of zero for $ $(x - 2)^3$ $?

    $3$. The exponent indicates how many times the zero appears.

12. What is the zero of a polynomial function?

    A value $x$ for which $f(x) = 0$. Values where the polynomial equals zero.

13. State the Fundamental Theorem of Algebra.

    Every non-zero polynomial has at least one complex root. This guarantees that all polynomial equations have solutions in the complex number system.

14. Find the sum of the roots of $x^2 - 5x + 6$.

    5. For $ax^2 + bx + c$, sum of roots equals $-b/a = 5/1$.

15. What is the degree of a polynomial?

    The highest power of the variable in the polynomial. Determines the polynomial's behavior and number of roots.

16. What is the degree of the zero polynomial?

    Undefined. The zero polynomial has no terms, so degree is undefined.

17. Calculate the modulus of $1 - i$.

    $\text{sqrt}(2)$. Use $|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}$.

18. What is the multiplicity of zero for $(x - 2)^3$?

    3. The exponent indicates how many times the zero appears.

19. What is the real part of $4 - 3i$?

    4. The coefficient of the term without $i$.

20. Identify the conjugate of $3 + 4i$.

    $3 - 4i$. Change the sign of the imaginary part to get the conjugate.

21. Define a complex number.

    A number in the form $a + bi$, where $a$ and $b$ are real numbers. Standard form where $i = \sqrt{-1}$ represents the imaginary unit.

22. What does the term 'roots' refer to in polynomials?

    Values that satisfy $f(x) = 0$. Same as zeros - values that make the polynomial equal zero.

23. What is the modulus of $3 + 4i$?

    5. Use $|a + bi| = \sqrt{a^2 + b^2} = \sqrt{9 + 16}$.

24. What is a polynomial's leading coefficient?

    Coefficient of the highest degree term. Same as leading coefficient - determines polynomial behavior.

25. What is the imaginary unit $i$?

    $i = \text{sqrt}(-1)$. Fundamental unit for complex numbers, where $i^2 = -1$.

26. Identify the zeros of $x^2 + 1$.

    $x = i, -i$. Set $x^2 = -1$ and solve for $x = \pm i$.

27. What is Descartes' Rule of Signs?

    Predicts the number of positive and negative real roots. Counts sign changes to estimate the number of positive/negative roots.

28. Find the product of $2 + i$ and $2 - i$.

    5. Conjugate pairs multiply to $a^2 + b^2 = 4 + 1 = 5$.

29. Calculate the discriminant of $x^2 + 4x + 5$.

    $-4$. Use $b^2 - 4ac = 16 - 20 = -4$ for negative discriminant.

30. What is the degree of the polynomial $0$?

    Undefined. The zero polynomial has no degree by convention.

31. Find the complex roots of $x^2 + 4$.

    $x = 2i, -2i$. Set $x^2 = -4$ and solve for $x = \pm 2i$.

32. Find the roots of $x^2 - 2x + 1$.

    $x = 1$. This is $(x-1)^2$, so $x = 1$ is a repeated root.

33. What is the standard form of a polynomial?

    Terms arranged in descending order of exponents. Conventional ordering from highest to lowest degree terms.

34. State the definition of a root of multiplicity.

    A root repeated in the factorization of a polynomial. Indicates how many times a root appears in the factorization.

35. What is the degree of $7x^3 + 2x^2 + x + 5$?

    3. The highest power of $x$ determines the degree.

36. State the meaning of 'complex conjugate'.

    Change the sign of the imaginary part of a complex number. Flip the sign of the imaginary part only.

37. What is the leading term of $2x^5 + 3x^4$?

    $2x^5$. The term with the highest degree in the polynomial.

38. What is a leading coefficient?

    The coefficient of the term with the highest degree. Determines the polynomial's end behavior and properties.

39. What is the imaginary part of $5 + 6i$?

    6. The coefficient of $i$ in the complex number.

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