Polynomial Equations: Degree & Terms

• Solves polynomial equations and classifies them by degree and number of terms.

Question 1

• (2x² + 4x + 12) + (3x² + 5x - 6) = 5x² + 9x + 6
• The answer, 5x² + 9x + 6, is a quadratic trinomial.

Question 2

• (4 - 5x⁵ - 7x⁴) - (-7x⁴ + 8x³ - 3x) = -5x⁵ - 7x⁴ + 7x⁴ - 8x³ + 3x + 4 = -5x⁵ + 8x³ + 3x + 4
• The answer, -5x⁵ + 8x³ + 3x + 4, is a 5th degree polynomial.

Question 3

• (-3x⁶ + 12 + 5x²) + (-5x⁶ - 2x² + 8) = -3x⁶ - 5x⁶ + 5x² - 2x² + 12 + 8 = -8x⁶ + 3x² + 20
• The answer, -8x⁶ + 3x² + 20, is a 6th degree trinomial.

Question 4

• (3x³ - 4x² + 12) - (2x³ + 2x² + 3x - 6) - (x³ + 8) = 3x³ - 2x³ - x³ - 4x² - 2x² - 3x + 12 + 6 - 8 = -6x² - 3x + 10
• The answer, -6x² - 3x + 10, is a quadratic trinomial.

Question 5

• 2x²(3x⁴ - 4x) = 6x⁶ - 8x³
• The answer, 6x⁶ - 8x³, is a 6th degree binomial.

Question 6

• (3x + 2)(-5x - 3) = -15x² - 9x - 10x - 6 = -15x² - 19x - 6.
• The answer, -15x² - 19x - 6, is a quadratic trinomial.

Question 7

• (2 - 3x)(3x² - 2x + 4) = 6x² - 4x + 8 - 9x³ + 6x² - 12x = -9x³ + 12x² - 16x + 8
• The answer, -9x³ + 12x² - 16x + 8, is a cubic polynomial.

Question 8

• (x - 3)(x + 2)(x - 1) = (x² + 2x - 3x - 6)(x - 1) = (x² - x - 6)(x - 1) = x³ - x² - x² + x - 6x + 6 = x³ - 2x² - 5x + 6
• The answer, x³ - 2x² - 5x + 6, is a cubic polynomial.

Question 9

• Calculates the area of the shaded region by subtracting the area of the inner rectangle from the area of the outer rectangle: [(x+4)(x+8)] - [(x+1)(x+3)]
• Expands and simplifies: [x² + 8x + 4x + 32] - [x² + 3x + x + 3] = [x² + 12x + 32] - [x² + 4x + 3] = 8x + 29.

Question 10

• Calculates the area of the shaded region, using [(2x+9)(x+6)] - [(x-5)(x-3)].
• Expands and simplifies: [2x² + 12x + 9x + 54] - [x² - 3x - 5x + 15] = [2x² + 21x + 54] - [x² - 8x + 15] = x² + 29x + 39.

Questions 11-16

• Factors the greatest common factor (GCF) out of each expression.

Question 11

• Factoring 8x out of 16x³/8x - 24x²/8x - 8x/8x results in 8x(2x² - 3x - 1).

Question 12

• Factoring -4 out of -16x²/-4 - 8x/-4 + 4/-4 results in -4(4x² + 2x - 1).

Question 13

• Factoring -5x² out of -5x⁴/-5x² + 40x³/-5x² - 10x²/-5x² results in -5x²(x² - 8x + 2).

Question 14

• Factoring 5 out of 25x⁵/5 + 5x³/5 - 20x/5 + 15/5 results in 5(5x³ + x² - 4x + 3).

Question 15

• Factoring 3x² out of 21x⁵/3x² - 84x⁴/3x² + 15x³/3x² - 60x/3x² results in 3x²(7x³ - 28x² + 5x - 20).

Question 16

• Factoring 8x out of 32x⁴/8x - 24x³/8x + 40x²/8x + 16x/8x results in 8x(4x³ - 3x² + 5x + 2).

Questions 17-24

• Factors the following trinomials.

Question 17

• Factors k² - 13k + 40 into (k - 8)(k - 5).

Question 18

• Factors x² + 2x - 24 into (x + 6)(x - 4).

Question 19

• Simplifies 2k² + 22k + 60 to 2(k² + 11k + 30), then factors it into 2(k + 6)(k + 5).

Question 20

• Simplifies n² + 10n - 9 to -(n² - 10n + 9), then factors it into -(n - 9)(n - 1).

Question 21

• Factors 3x² - 2x - 5 into (3x - 5)(x + 1).

Question 22

• Factors 2x² + 3x - 9 into (x + 3)(2x - 3).

Question 23

• Simplifies 15x² - 27x - 6 to 3(5x² - 9x - 2), then attempts to factors it into 3(x - 2)(5x + 1)

Question 24

• Factors 5x³ - 18x² + 9x into x(5x² - 18x + 9), then factors it into x(x - 3)(5x - 3).