Understanding and Operating with Polynomials - Algebra 2

1. What is the degree of the constant monomial $-9$?

   $0$. Constants have degree zero by definition.

2. Find and correct the error: $(x+3)(x+4)=x^2+7x+7$.

   Correct: $(x+3)(x+4)=x^2+7x+12$. The constant term should be $3 \times 4 = 12$.

3. Find and correct the error: $(2x)(3x^2-1)=6x^2-2x$.

   Correct: $(2x)(3x^2-1)=6x^3-2x$. Power rule: $x \times x^2 = x^3$, not $x^2$.

4. What is the standard form of a polynomial written in one variable $x$?

   Terms in descending powers of $x$ with like terms combined. Highest to lowest powers with simplified terms.

5. What is the leading term of the polynomial $4x^3-2x+7$?

   $4x^3$. The term with the highest degree comes first.

6. What is the product $(x+3)^2$ expanded?

   $x^2+6x+9$. Square the binomial: $(a+b)^2=a^2+2ab+b^2$.

7. What is the product $3x(2x^2-5x+4)$?

   $6x^3-15x^2+12x$. Distribute the monomial to each term.

8. What is the result of $ ($6x^2-1$) - ($2x^2+5$)$?

   $4x^2-6$. Distribute negative and combine like terms.

9. What is the result of $-(3x^2-4x+6)$ written as a polynomial?

   $-3x^2+4x-6$. Distribute the negative sign to all terms.

10. What is the result of $ ($x^2+3x+1$) + ($-x^2$+4x-6)$?

    $7x-5$. Add corresponding coefficients of like terms.

11. Identify the simplified form of $(x^2-3x+1)-(2x^2+x-5)$.

    $-x^2-4x+6$. Distribute negative and combine like terms.

12. What is the product $-2x^2(3x^3-x+6)$?

    $-6x^5+2x^3-12x^2$. Distribute the monomial to each term.

13. What is the product $(x+5)(x+2)$?

    $x^2+7x+10$. Use FOIL method to multiply binomials.

14. What is the product $(2x+1)(x^2-x+3)$?

    $2x^3-x^2+5x+3$. Distribute binomial to each term in the polynomial.

15. What is the product $(x-1)(x^2+2x+3)$?

    $x^3+x^2+x-3$. Distribute binomial to each term in the polynomial.

16. What is the result of subtracting $ ($2x^3+x-9$) - ($5x^3-4x+1$)$?

    $-3x^3+5x-10$. Distribute the negative and combine like terms.

17. What is the leading coefficient of the polynomial $-3x^4+5x^2-1$?

    $-3$. Coefficient of the highest degree term.

18. What is the result of multiplying $(5)(2x^2-x+3)$?

    $10x^2-5x+15$. Distribute the scalar to each term.

19. What is the result of combining like terms: $-x^3+4x^2+7-x^3-2x^2$?

    $-2x^3+2x^2+7$. Group and add coefficients of like terms.

20. What is the product $(x^2+1)(x^2-1)$?

    $x^4-1$. Use difference of squares formula: $a^2-b^2$.

21. What is the product $(x-2)^2$ expanded?

    $x^2-4x+4$. Square the binomial: $(a-b)^2=a^2-2ab+b^2$.

22. What is the result of adding $ ($3x^2+5x-1$) + ($2x^2-3x+4$)$?

    $5x^2+2x+3$. Add coefficients of like terms separately.

23. What is the result of multiplying $ (5)($2x^2-x+3$) $?

    $10x^2-5x+15$. Distribute the scalar to each term.

24. What is the product $(x^2-4)(x+1)$?

    $x^3+x^2-4x-4$. Distribute first polynomial to each term in second.

25. What is the degree of the monomial $7x^5$?

    $5$. The degree is the exponent of the variable.

26. What is a like term in a polynomial expression?

    A term with the same variable part, such as $x^3$ terms. Same variables raised to the same powers.

27. Identify the polynomial degree: $x^7+3x^2-8x+1$.

    $7$. Highest exponent determines the degree.

28. What is the degree of the polynomial $6x^2-4x+9$?

    $2$. Highest exponent determines the degree.

29. Find and correct the error: $(x-5)^2=x^2-25$.

    Correct: $(x-5)^2=x^2-10x+25$. Missing middle term: $(a-b)^2=a^2-2ab+b^2$.

30. What does it mean that polynomials are closed under multiplication?

    The product of two polynomials is always a polynomial. The result stays within the polynomial system.

31. What is the product $(2x-3)(x+5)$?

    $2x^2+7x-15$. Use FOIL method to multiply binomials.

32. What is the result of subtracting $ ($4x^2-6x+1$) - ($x^2+2x-3$)$?

    $3x^2-8x+4$. Distribute the negative and combine like terms.

33. What is the product $(x-4)(x+6)$?

    $x^2+2x-24$. Use FOIL method to multiply binomials.

34. What is the product $(x+2)(x^2+3x+4)$?

    $x^3+5x^2+10x+8$. Distribute binomial to each term in the polynomial.

35. What is the result of adding $ ($7x^3-2x$) + ($-7x^3$+9x+5)$?

    $7x+5$. The cubic terms cancel, leaving linear and constant.

36. What is the product $(x^2+2x)(x-3)$?

    $x^3-x^2-6x$. Distribute first polynomial to each term in second.

37. What does it mean that polynomials are closed under subtraction?

    The difference of two polynomials is always a polynomial. The result stays within the polynomial system.

38. Identify the simplified form of $ ($x^2-3x+1$)-($2x^2+x-5$) $.

    $ -$x^2$-4x+6 $. Distribute negative and combine like terms.

39. What does it mean that polynomials are closed under addition?

    The sum of two polynomials is always a polynomial. The result stays within the polynomial system.

40. What is the result of multiplying $(-x)(x^2-3x+2)$?

    $-x^3+3x^2-2x$. Distribute the negative monomial to each term.

41. What is the result of combining like terms: $5x^2-3x+2x^2+9x-4$?

    $7x^2+6x-4$. Group and add coefficients of like terms.

42. What is the product $(3x+1)(2x-5)$?

    $6x^2-13x-5$. Use FOIL method to multiply binomials.

43. Which expression is the additive inverse of the polynomial $4x^2-7x+9$?

    $-4x^2+7x-9$. Negate all coefficients to find additive inverse.

44. Find the product $(x+1)(x^2-1)$ in standard form.

    $x^3+x^2-x-1$. Expand $(x+1)(x+1)(x-1)$ step by step.

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