Activities

1. Rewrite the following polynomial in standard form. f(x) = x - 4 + 3x³ - 2x⁴ + 5x²

2. What is the degree, leading coefficient and constant term of the following polynomial?
   f(x) = x – 4 + 3x³ – 2x⁴ + 5x²

3. Rewrite the following polynomial in standard form. f(x) = 3x – 1 + 4x – 7

4. Rewrite the following polynomial in standard form. f(x) = 3x – (1 + 4x – 7)

5. Simplify the following polynomial expression.  (4x² + 7x – 5) + (3x – 1 – 9x²)

6. Simplify the following polynomial expression.  (4a – 3a³ + 2a) – (6a² – a³ – 11a + 2)

7. Simplify the following polynomial expression.  (3x – 4) (2x + 5)

8. Simplify the following polynomial expression. (5m² – 7m – 1)(2m + 3)

9. Simplify the following polynomial expression.  2x(x – 1) – x(9 + x)

10. Simplify the following polynomial expression. ((3x + 2) – (x – 1))(4 – 2x)

11. Simplify the following polynomial expression.  (x² – y + 3x)(2y – 4x)

12. A polynomial is shown below.

(kx – 7)(3x + 2)

For what value of k does the above polynomial expression equal 15x² – 11x – 14?

13. Given the equation (4x – 8) – p = -x + 4 where p can be written in terms of x, what is p?

14. What is the product of ax + b and cx + d?

15. A rectangle with a length of 3x – 4 is shown below.

The perimeter of the rectangle is -16 + 10x, what is the width of the rectangle?

16. A rectangular prism has height of 2x – 1 feet, a width of 4 – x feet, and a length of 3x feet. What is the volume of the prism in terms of x?

17. Samantha’s front porch measures 5x + 1 feet by 4x – 3 feet. A small carpet outside the front door has an area of 2x + 1 feet squared. What is the area of Samantha’s front porch that is not carpeted?

18. An equation is shown below.

(3x² – 4x + 9)(mx + 5) – (9x³ + 4x² + x – 11) = 6x³ – 9x² + 24x + 56

For what value of m is this equation true?

19. Jhamir is trying to solve an equation of the form 2a + b = c, where a, b, and c are all polynomials of the 2^nd degree. What could polynomial a and b equal so that c would be the third degree polynomial, 

2x³ – x² + 5x – 1.

Answer Key/Rubric

1. f(x) = -2x⁴ + 3x³ + 5x² + x – 4
2. Degree: 4 (Quartic), Leading coefficient: -2, Constant: -4
3. f(x) = 7x – 8
4. f(x) = -x + 6
5. -5x² + 10x – 6
6. -2a³ – 6a² + 17a – 2
7. 6x² + 7x – 20
8. 10m³ + m² – 23m – 3  
9. x² – 11x
10. = -4x² + 2x + 12
11. -4x³ – 12x² + 2x²y + 10xy – 2y²
12. k = 5
13. p = 5x – 12
14. acx² + adx + bcx + bd
15. width = 2x – 4
16. Volume = -6x³ + 27x² – 12x
17. 20x² – 13x – 4
18. m = 5
19. Various answers: could be a = 2x³ + x² + 3x + 1 and b = -2x³ – 3x² – x – 3