Polynomials Practice Problems Review Notes

Polynomials

Degree of Polynomials

• Question 1: Find the degree of the polynomial: $x^3 + 2x - 3x^4 + 5 + 3x^2$.
  • The degree of a polynomial is the highest power of the variable in the polynomial.
  • In this case, the highest power of $x$ is 4 (from the term $-3x^4$).
  • Therefore, the degree of the polynomial is 4.
• Question 2: Find the degree of the polynomial: $3 - 14x^4 - 2x^8 + 20x + 4x^2$.
  • The highest power of $x$ in this polynomial is 8 (from the term $-2x^8$).
  • Therefore, the degree of the polynomial is 8.
• Question 3: What is the coefficient of the term of degree 6 in the polynomial $3x^2 - 14x^6 + 6x^3 + 12x + 5x^5$?
  • The term with degree 6 is $-14x^6$.
  • The coefficient of this term is -14.
  • Therefore, the coefficient of the term of degree 6 is -14.

Sum and Difference of Polynomials

• Question 4: Find the sum of the polynomials: $(9x^2 + 3x - 2) + (3x^2 - 5x - 3)$.
  • Combine like terms: $(9x^2 + 3x^2) + (3x - 5x) + (-2 - 3)$.
  • This simplifies to $12x^2 - 2x - 5$.
• Question 5: Find the difference of the polynomials: $(3x^2 - 3x + 6) - (5x^2 + 2x + 9)$.
  • Distribute the negative sign and combine like terms: $3x^2 - 3x + 6 - 5x^2 - 2x - 9$.
  • Combine like terms: $(3x^2 - 5x^2) + (-3x - 2x) + (6 - 9)$.
  • This simplifies to $-2x^2 - 5x - 3$.
• Question 6: Find the sum of the polynomials: $(-3x^2 + 2x + 1) + (-8x^2 + 3x + 2)$.
  • Combine like terms: $(-3x^2 - 8x^2) + (2x + 3x) + (1 + 2)$.
  • This simplifies to $-11x^2 + 5x + 3$.
• Question 7: Find the difference of the polynomials: $(2x^2 + 5x - 12) - (-4x^2 + 2x + 6)$.
  • Distribute the negative sign and combine like terms: $2x^2 + 5x - 12 + 4x^2 - 2x - 6$.
  • Combine like terms: $(2x^2 + 4x^2) + (5x - 2x) + (-12 - 6)$.
  • This simplifies to $6x^2 + 3x - 18$.
• Question 8: Find the sum of the polynomials: $(3x^2 + 7x + 4) + (7x^2 - 7x + 2)$.
  • Combine like terms: $(3x^2 + 7x^2) + (7x - 7x) + (4 + 2)$.
  • This simplifies to $10x^2 + 6$.
• Question 9: Find the difference of the polynomials: $(-2x^2 - 3x - 3) - (9x - 3)$.
  • Distribute the negative sign and combine like terms: $-2x^2 - 3x - 3 - 9x + 3$.
  • Combine like terms: $-2x^2 + (-3x - 9x) + (-3 + 3)$.
  • This simplifies to $-2x^2 - 12x$.

Polynomial Multiplication Table

• Given a table set up to multiply two polynomials, which includes the terms $x^3, -2x^2, 5$ for one polynomial and $3x, 8$ for the other.
• Question 10: What is the coefficient of the $x^3$-term of the product?
  • We need to determine how the $x^3$ term is formed in the product.
  • From the table the coefficient of the $x^3$ is 8.
• Question 11: What is the coefficient of the $x^2$-term of the product?
  • We need to determine how the $x^2$ term is formed in the product.
  • From the table the coefficient of the $x^2$ is -6.

Product of Two Polynomials

• Question 12: Find the product of the two polynomials: $(2x + 7)(2x - 7)$.
  • Using the difference of squares formula: $(a + b)(a - b) = a^2 - b^2$ where $a = 2x$ and $b = 7$.
  • $(2x + 7)(2x - 7) = (2x)^2 - (7)^2 = 4x^2 - 49$.
• Question 13: Find the product of the two polynomials: $(8x^2 - 2x + 2)(3x + 5)$.
  • Distribute each term: $8x^2(3x + 5) - 2x(3x + 5) + 2(3x + 5)$.
  • $= 24x^3 + 40x^2 - 6x^2 - 10x + 6x + 10$.
  • Combine like terms: $24x^3 + (40x^2 - 6x^2) + (-10x + 6x) + 10$.
  • $= 24x^3 + 34x^2 - 4x + 10$.
• Question 14: Find the product of the two polynomials: $(2x - 3)(x^2 + 4x + 1)$.
  • Distribute each term: $2x(x^2 + 4x + 1) - 3(x^2 + 4x + 1)$.
  • $= 2x^3 + 8x^2 + 2x - 3x^2 - 12x - 3$.
  • Combine like terms: $2x^3 + (8x^2 - 3x^2) + (2x - 12x) - 3$.
  • $= 2x^3 + 5x^2 - 10x - 3$