Equivalent Polynomial and Rational Expressions - AP Precalculus

1. What is the standard form of a quadratic polynomial?

   $ax^2 + bx + c$. The general form where $a \neq 0$.

2. What is the quotient of $\frac{x^2 - 4}{x - 2}$?

   $x + 2$. Factor numerator: $\frac{(x+2)(x-2)}{x-2} = x+2$.

3. What is the expression for $(x - 3)^2$?

   $x^2 - 6x + 9$. Use $(a-b)^2 = a^2 - 2ab + b^2$.

4. What is the sum of the roots of $x^2 - 4x + 4$?

   4. For $ax^2 + bx + c$, sum of roots is $-\frac{b}{a}$.

5. Rewrite $\frac{x^2 + 5x + 6}{x + 2}$ in simplest form.

   $x + 3$. Factor and cancel: $\frac{(x+2)(x+3)}{x+2} = x+3$.

6. Which term refers to the highest degree term in a polynomial?

   Leading term. The term with the highest degree in standard form.

7. Identify the expression equivalent to $\frac{x^3 - x}{x}$.

   $x^2 - 1$. Factor and cancel: $\frac{x(x^2-1)}{x} = x^2-1$.

8. Simplify $\frac{x^2 - 3x}{x - 3}$.

   $x$. Factor numerator: $\frac{x(x-3)}{x-3} = x$.

9. What is the degree of the polynomial $4x^3 - 2x + 7$?

   3. The highest power of $x$ in the polynomial.

10. What is the standard form of a linear polynomial?

    $ax + b$. The general form of a first-degree polynomial.

11. Find the product of $(x + 1)$ and $(x - 1)$.

    $x^2 - 1$. Difference of squares formula gives $x^2 - 1^2$.

12. Identify the expression equivalent to $\frac{x^2 + 2x}{x}$.

    $x + 2$. Factor out $x$: $\frac{x(x+2)}{x} = x+2$.

13. State the discriminant of the quadratic $x^2 + 4x + 4$.

    0. For perfect square, discriminant $b^2 - 4ac = 0$.

14. Convert $\frac{3x + 6}{x + 2}$ to simplest form.

    3. Factor and cancel: $\frac{3(x+2)}{x+2} = 3$.

15. What is the expression for the cube of $(x + 2)$?

    $x^3 + 6x^2 + 12x + 8$. Use $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

16. Simplify the expression $\frac{x^2 - 4x}{x}$.

    $x - 4$. Factor out $x$: $\frac{x(x-4)}{x} = x-4$.

17. What is the binomial theorem?

    $(a + b)^n = \text{sum of terms } C(n, k)a^{n-k}b^k$. Expands $ $(a + b)^n$ $ using combinations and powers.

18. What is the factored form of $x^2 - 9$?

    $(x - 3)(x + 3)$. Difference of squares: $a^2 - b^2 = (a-b)(a+b)$.

19. Identify the leading coefficient in $5x^4 - 3x^2 + x$.

    5. The coefficient of the highest degree term.

20. What is the expression for the sum of $2x^2 + 3x$ and $x^2 - x$?

    $3x^2 + 2x$. Combine like terms: $(2+1)x^2 + (3-1)x = 3x^2 + 2x$.

21. Simplify the expression $2x(x^2 - 3x + 4)$.

    $2x^3 - 6x^2 + 8x$. Distribute $2x$ to each term inside parentheses.

22. What does it mean for a polynomial to be monic?

    Leading coefficient is 1. The coefficient of the highest degree term equals 1.

23. Factor $x^2 + 6x + 9$.

    $(x + 3)^2$. Perfect square trinomial: $a^2 + 2ab + b^2 = (a+b)^2$.

24. What is the expression for the difference between $x^2 + 4x$ and $2x^2 + x$?

    $-x^2 + 3x$. $(x^2 + 4x) - (2x^2 + x) = -x^2 + 3x$.

25. State the expression for the difference of cubes $a^3 - b^3$.

    $(a - b)(a^2 + ab + b^2)$. The factorization formula for $a^3 - b^3$.

26. What is the polynomial remainder theorem?

    Remainder of $f(x)$ divided by $x - a$ is $f(a)$. Substitute $x=a$ into $f(x)$ to find remainder.

27. Express $3x^2 - 12$ as a product of its factors.

    $3(x^2 - 4)$. Factor out common factor 3 first.

28. State the remainder when $x^3 + 3x^2 - x + 5$ is divided by $x - 1$.

    8. By remainder theorem, substitute $x=1$: $1+3-1+5=8$.

29. What is the product of the roots of $x^2 - 5x + 6$?

    6. For $ax^2 + bx + c$, product of roots is $\frac{c}{a}$.

30. What is the term for a polynomial with three terms?

    Trinomial. A polynomial with exactly three terms.

31. What is the expression for the square of $(x + 2)$?

    $x^2 + 4x + 4$. Use $(a+b)^2 = a^2 + 2ab + b^2$.

32. Simplify the expression $\frac{x^2 - 9}{x + 3}$.

    $x - 3$. Factor numerator: $\frac{(x+3)(x-3)}{x+3} = x-3$.

33. What is the greatest common factor of $6x^2 + 9x$?

    3x. Factor out the GCF: $3x(2x + 3)$.

34. Express $x^3 - 27$ as a product of factors.

    $(x - 3)(x^2 + 3x + 9)$. Difference of cubes: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$.

35. What is the common denominator of $\frac{1}{x}$ and $\frac{1}{x^2}$?

    $x^2$. Use the highest power of $x$ in both denominators.

36. What is the expanded form of $(x + 1)^3$?

    $x^3 + 3x^2 + 3x + 1$. Use $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.

37. State the zeroes of the polynomial $x^2 - 5x + 6$.

    2, 3. Factor as $(x-2)(x-3)$ and set each factor to zero.

38. Convert $f(x) = (x - 2)(x + 3)$ to standard form.

    $x^2 + x - 6$. Use FOIL: $(x-2)(x+3) = x^2 + 3x - 2x - 6$.

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