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Step-by-step solution for: Polynomials Worksheet #1 | Study Guides, Projects, Research ...
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Step-by-step solution for: Polynomials Worksheet #1 | Study Guides, Projects, Research ...
Let's go through each part of the worksheet step by step and solve all the problems with clear explanations.
---
Definitions:
- Monomial (M): One term (e.g., $5x$, $-3$)
- Binomial (B): Two terms (e.g., $x + 1$)
- Trinomial (T): Three terms (e.g., $x^2 + 2x + 1$)
- Polynomial (P): Any expression with one or more terms (includes monomials, binomials, trinomials) — but usually used for expressions with more than three terms or general classification.
- Constant (C): A number without variables (e.g., $-130$)
Now classify:
1.) $2x + 1$ → Two terms → B (binomial)
2.) $17x^2 + 11$ → Two terms → B (binomial)
3.) $8x^3 + 2x^2 + 3x - 7$ → Four terms → P (polynomial)
4.) $-130$ → No variable, just a number → C (constant)
5.) $4a^2 + 7a - 10$ → Three terms → T (trinomial)
6.) $10x^3 - 2x + 1$ → Three terms → T (trinomial)
✔ Answers:
1. B
2. B
3. P
4. C
5. T
6. T
---
Use PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
7.) $4 + 2 \times 12 \div 6 - 8$
→ First: $2 \times 12 = 24$
→ Then: $24 \div 6 = 4$
→ Now: $4 + 4 - 8 = 0$
✔ Answer: 0
8.) $9 \div 3 + 6 \times 2$
→ $9 \div 3 = 3$
→ $6 \times 2 = 12$
→ $3 + 12 = 15$
✔ Answer: 15
9.) $3 + 4 - 6 + 2 \times 35$
→ $2 \times 35 = 70$
→ Now: $3 + 4 - 6 + 70 = (7 - 6) + 70 = 1 + 70 = 71$
✔ Answer: 71
10.) $\frac{2}{7} + \frac{3}{7}$ → Same denominator: $\frac{2+3}{7} = \frac{5}{7}$
✔ Answer: $\frac{5}{7}$
11.) $\frac{4}{5} + \frac{1}{2}$ → Find common denominator: 10
→ $\frac{4}{5} = \frac{8}{10},\ \frac{1}{2} = \frac{5}{10}$
→ $\frac{8}{10} + \frac{5}{10} = \frac{13}{10} = 1\frac{3}{10}$
✔ Answer: $\frac{13}{10}$ or $1\frac{3}{10}$
12.) $\frac{3}{5} - \frac{1}{3}$ → LCD = 15
→ $\frac{3}{5} = \frac{9}{15},\ \frac{1}{3} = \frac{5}{15}$
→ $\frac{9}{15} - \frac{5}{15} = \frac{4}{15}$
✔ Answer: $\frac{4}{15}$
---
Like terms: Same variable and exponent.
12.) $(19x^2 + 12x + 12) + (7x^2 + 10x + 13)$
→ $19x^2 + 7x^2 = 26x^2$
→ $12x + 10x = 22x$
→ $12 + 13 = 25$
✔ Answer: $26x^2 + 22x + 25$
13.) $(4x^2 - 6x + 7) + (-19x^2 - 15x - 18)$
→ $4x^2 - 19x^2 = -15x^2$
→ $-6x - 15x = -21x$
→ $7 - 18 = -11$
✔ Answer: $-15x^2 - 21x - 11$
14.) $(20x^2 + 15x + 13) + (-19x^2 + 17x + 5)$
→ $20x^2 - 19x^2 = x^2$
→ $15x + 17x = 32x$
→ $13 + 5 = 18$
✔ Answer: $x^2 + 32x + 18$
15.) $(9x^6 - 4x^5) + (10x^5 - 15x^4 + 14)$
→ $9x^6$ (no other $x^6$)
→ $-4x^5 + 10x^5 = 6x^5$
→ $-15x^4$ (only one)
→ $+14$ (constant)
✔ Answer: $9x^6 + 6x^5 - 15x^4 + 14$
---
Subtracting means distribute the negative sign first.
18.) $(6x + 14) - (9x + 5)$
→ $6x + 14 - 9x - 5 = (6x - 9x) + (14 - 5) = -3x + 9$
✔ Answer: $-3x + 9$
20.) $(19x^2 + 9x + 16) - (5x^2 + 12x + 7)$
→ $19x^2 - 5x^2 = 14x^2$
→ $9x - 12x = -3x$
→ $16 - 7 = 9$
✔ Answer: $14x^2 - 3x + 9$
21.) $(17x^2 + 7x - 14) - (-6x^2 - 5x - 18)$
→ Distribute the minus: $+6x^2 + 5x + 18$
→ $17x^2 + 6x^2 = 23x^2$
→ $7x + 5x = 12x$
→ $-14 + 18 = 4$
✔ Answer: $23x^2 + 12x + 4$
22.) $(-18x^2 + 4x - 16) - (15x^2 + 4x - 1)$
→ $-18x^2 - 15x^2 = -33x^2$
→ $4x - 4x = 0$
→ $-16 - (-1) = -16 + 1 = -15$
✔ Answer: $-33x^2 - 15$
---
Distribute the outside term to each inside term.
29.) $6(x^2 + 2x + 7)$
→ $6 \cdot x^2 = 6x^2$
→ $6 \cdot 2x = 12x$
→ $6 \cdot 7 = 42$
✔ Answer: $6x^2 + 12x + 42$
30.) $4x(1 - x)$
→ $4x \cdot 1 = 4x$
→ $4x \cdot (-x) = -4x^2$
→ $-4x^2 + 4x$
✔ Answer: $-4x^2 + 4x$
30.) $-x^2(x + 5)$
→ $-x^2 \cdot x = -x^3$
→ $-x^2 \cdot 5 = -5x^2$
✔ Answer: $-x^3 - 5x^2$
31.) $3x^2(4x^3 - 5x + 10)$
→ $3x^2 \cdot 4x^3 = 12x^5$
→ $3x^2 \cdot (-5x) = -15x^3$
→ $3x^2 \cdot 10 = 30x^2$
✔ Answer: $12x^5 - 15x^3 + 30x^2$
32.) $3x(-x^2 + 2x - 12)$
→ $3x \cdot (-x^2) = -3x^3$
→ $3x \cdot 2x = 6x^2$
→ $3x \cdot (-12) = -36x$
✔ Answer: $-3x^3 + 6x^2 - 36x$
---
33.) $(x - 3)(x + 4)$
→ FOIL:
First: $x \cdot x = x^2$
Outer: $x \cdot 4 = 4x$
Inner: $-3 \cdot x = -3x$
Last: $-3 \cdot 4 = -12$
→ $x^2 + 4x - 3x - 12 = x^2 + x - 12$
✔ Answer: $x^2 + x - 12$
34.) $(2x + 4)(2x + 3)$
→ First: $2x \cdot 2x = 4x^2$
→ Outer: $2x \cdot 3 = 6x$
→ Inner: $4 \cdot 2x = 8x$
→ Last: $4 \cdot 3 = 12$
→ $4x^2 + 6x + 8x + 12 = 4x^2 + 14x + 12$
✔ Answer: $4x^2 + 14x + 12$
35.) $(x - 7)(x - 6)$
→ $x \cdot x = x^2$
→ $x \cdot (-6) = -6x$
→ $-7 \cdot x = -7x$
→ $-7 \cdot (-6) = 42$
→ $x^2 - 6x - 7x + 42 = x^2 - 13x + 42$
✔ Answer: $x^2 - 13x + 42$
36.) $(3x - 1)(x + 5)$
→ $3x \cdot x = 3x^2$
→ $3x \cdot 5 = 15x$
→ $-1 \cdot x = -x$
→ $-1 \cdot 5 = -5$
→ $3x^2 + 15x - x - 5 = 3x^2 + 14x - 5$
✔ Answer: $3x^2 + 14x - 5$
37.) $(x + 5)(x^2 - 6x + 3)$
→ Distribute each term in first binomial:
→ $x(x^2 - 6x + 3) = x^3 - 6x^2 + 3x$
→ $5(x^2 - 6x + 3) = 5x^2 - 30x + 15$
→ Add:
→ $x^3 - 6x^2 + 3x + 5x^2 - 30x + 15$
→ $x^3 + (-6x^2 + 5x^2) + (3x - 30x) + 15 = x^3 - x^2 - 27x + 15$
✔ Answer: $x^3 - x^2 - 27x + 15$
38.) $(2x - 3)(4x^2 + 8x - 2)$
→ $2x(4x^2 + 8x - 2) = 8x^3 + 16x^2 - 4x$
→ $-3(4x^2 + 8x - 2) = -12x^2 - 24x + 6$
→ Add:
→ $8x^3 + 16x^2 - 4x - 12x^2 - 24x + 6$
→ $8x^3 + (16x^2 - 12x^2) + (-4x - 24x) + 6 = 8x^3 + 4x^2 - 28x + 6$
✔ Answer: $8x^3 + 4x^2 - 28x + 6$
39.) $(2x - 1)^2 = (2x - 1)(2x - 1)$
→ $2x \cdot 2x = 4x^2$
→ $2x \cdot (-1) = -2x$
→ $-1 \cdot 2x = -2x$
→ $-1 \cdot (-1) = 1$
→ $4x^2 - 2x - 2x + 1 = 4x^2 - 4x + 1$
✔ Answer: $4x^2 - 4x + 1$
40.) $(4x + 3)^2 = (4x + 3)(4x + 3)$
→ $4x \cdot 4x = 16x^2$
→ $4x \cdot 3 = 12x$
→ $3 \cdot 4x = 12x$
→ $3 \cdot 3 = 9$
→ $16x^2 + 12x + 12x + 9 = 16x^2 + 24x + 9$
✔ Answer: $16x^2 + 24x + 9$
---
#### Part 1:
1. B
2. B
3. P
4. C
5. T
6. T
#### Part 2:
7. 0
8. 15
9. 71
10. $\frac{5}{7}$
11. $\frac{13}{10}$ or $1\frac{3}{10}$
12. $\frac{4}{15}$
#### Part 3:
12. $26x^2 + 22x + 25$
13. $-15x^2 - 21x - 11$
14. $x^2 + 32x + 18$
15. $9x^6 + 6x^5 - 15x^4 + 14$
#### Part 4:
18. $-3x + 9$
20. $14x^2 - 3x + 9$
21. $23x^2 + 12x + 4$
22. $-33x^2 - 15$
#### Part 5:
29. $6x^2 + 12x + 42$
30. $-4x^2 + 4x$
30. $-x^3 - 5x^2$
31. $12x^5 - 15x^3 + 30x^2$
32. $-3x^3 + 6x^2 - 36x$
#### Part 6:
33. $x^2 + x - 12$
34. $4x^2 + 14x + 12$
35. $x^2 - 13x + 42$
36. $3x^2 + 14x - 5$
37. $x^3 - x^2 - 27x + 15$
38. $8x^3 + 4x^2 - 28x + 6$
39. $4x^2 - 4x + 1$
40. $16x^2 + 24x + 9$
Let me know if you'd like this formatted as a printable answer key!
---
Part 1: Classify each as M (monomial), B (binomial), T (trinomial), P (polynomial), or C (constant).
Definitions:
- Monomial (M): One term (e.g., $5x$, $-3$)
- Binomial (B): Two terms (e.g., $x + 1$)
- Trinomial (T): Three terms (e.g., $x^2 + 2x + 1$)
- Polynomial (P): Any expression with one or more terms (includes monomials, binomials, trinomials) — but usually used for expressions with more than three terms or general classification.
- Constant (C): A number without variables (e.g., $-130$)
Now classify:
1.) $2x + 1$ → Two terms → B (binomial)
2.) $17x^2 + 11$ → Two terms → B (binomial)
3.) $8x^3 + 2x^2 + 3x - 7$ → Four terms → P (polynomial)
4.) $-130$ → No variable, just a number → C (constant)
5.) $4a^2 + 7a - 10$ → Three terms → T (trinomial)
6.) $10x^3 - 2x + 1$ → Three terms → T (trinomial)
✔ Answers:
1. B
2. B
3. P
4. C
5. T
6. T
---
Part 2: Order of Operations and Fractions
Use PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
7.) $4 + 2 \times 12 \div 6 - 8$
→ First: $2 \times 12 = 24$
→ Then: $24 \div 6 = 4$
→ Now: $4 + 4 - 8 = 0$
✔ Answer: 0
8.) $9 \div 3 + 6 \times 2$
→ $9 \div 3 = 3$
→ $6 \times 2 = 12$
→ $3 + 12 = 15$
✔ Answer: 15
9.) $3 + 4 - 6 + 2 \times 35$
→ $2 \times 35 = 70$
→ Now: $3 + 4 - 6 + 70 = (7 - 6) + 70 = 1 + 70 = 71$
✔ Answer: 71
10.) $\frac{2}{7} + \frac{3}{7}$ → Same denominator: $\frac{2+3}{7} = \frac{5}{7}$
✔ Answer: $\frac{5}{7}$
11.) $\frac{4}{5} + \frac{1}{2}$ → Find common denominator: 10
→ $\frac{4}{5} = \frac{8}{10},\ \frac{1}{2} = \frac{5}{10}$
→ $\frac{8}{10} + \frac{5}{10} = \frac{13}{10} = 1\frac{3}{10}$
✔ Answer: $\frac{13}{10}$ or $1\frac{3}{10}$
12.) $\frac{3}{5} - \frac{1}{3}$ → LCD = 15
→ $\frac{3}{5} = \frac{9}{15},\ \frac{1}{3} = \frac{5}{15}$
→ $\frac{9}{15} - \frac{5}{15} = \frac{4}{15}$
✔ Answer: $\frac{4}{15}$
---
Part 3: Add these polynomials. Combine like terms.
Like terms: Same variable and exponent.
12.) $(19x^2 + 12x + 12) + (7x^2 + 10x + 13)$
→ $19x^2 + 7x^2 = 26x^2$
→ $12x + 10x = 22x$
→ $12 + 13 = 25$
✔ Answer: $26x^2 + 22x + 25$
13.) $(4x^2 - 6x + 7) + (-19x^2 - 15x - 18)$
→ $4x^2 - 19x^2 = -15x^2$
→ $-6x - 15x = -21x$
→ $7 - 18 = -11$
✔ Answer: $-15x^2 - 21x - 11$
14.) $(20x^2 + 15x + 13) + (-19x^2 + 17x + 5)$
→ $20x^2 - 19x^2 = x^2$
→ $15x + 17x = 32x$
→ $13 + 5 = 18$
✔ Answer: $x^2 + 32x + 18$
15.) $(9x^6 - 4x^5) + (10x^5 - 15x^4 + 14)$
→ $9x^6$ (no other $x^6$)
→ $-4x^5 + 10x^5 = 6x^5$
→ $-15x^4$ (only one)
→ $+14$ (constant)
✔ Answer: $9x^6 + 6x^5 - 15x^4 + 14$
---
Part 4: Subtract these polynomials
Subtracting means distribute the negative sign first.
18.) $(6x + 14) - (9x + 5)$
→ $6x + 14 - 9x - 5 = (6x - 9x) + (14 - 5) = -3x + 9$
✔ Answer: $-3x + 9$
20.) $(19x^2 + 9x + 16) - (5x^2 + 12x + 7)$
→ $19x^2 - 5x^2 = 14x^2$
→ $9x - 12x = -3x$
→ $16 - 7 = 9$
✔ Answer: $14x^2 - 3x + 9$
21.) $(17x^2 + 7x - 14) - (-6x^2 - 5x - 18)$
→ Distribute the minus: $+6x^2 + 5x + 18$
→ $17x^2 + 6x^2 = 23x^2$
→ $7x + 5x = 12x$
→ $-14 + 18 = 4$
✔ Answer: $23x^2 + 12x + 4$
22.) $(-18x^2 + 4x - 16) - (15x^2 + 4x - 1)$
→ $-18x^2 - 15x^2 = -33x^2$
→ $4x - 4x = 0$
→ $-16 - (-1) = -16 + 1 = -15$
✔ Answer: $-33x^2 - 15$
---
Part 5: Use distributive property to multiply
Distribute the outside term to each inside term.
29.) $6(x^2 + 2x + 7)$
→ $6 \cdot x^2 = 6x^2$
→ $6 \cdot 2x = 12x$
→ $6 \cdot 7 = 42$
✔ Answer: $6x^2 + 12x + 42$
30.) $4x(1 - x)$
→ $4x \cdot 1 = 4x$
→ $4x \cdot (-x) = -4x^2$
→ $-4x^2 + 4x$
✔ Answer: $-4x^2 + 4x$
30.) $-x^2(x + 5)$
→ $-x^2 \cdot x = -x^3$
→ $-x^2 \cdot 5 = -5x^2$
✔ Answer: $-x^3 - 5x^2$
31.) $3x^2(4x^3 - 5x + 10)$
→ $3x^2 \cdot 4x^3 = 12x^5$
→ $3x^2 \cdot (-5x) = -15x^3$
→ $3x^2 \cdot 10 = 30x^2$
✔ Answer: $12x^5 - 15x^3 + 30x^2$
32.) $3x(-x^2 + 2x - 12)$
→ $3x \cdot (-x^2) = -3x^3$
→ $3x \cdot 2x = 6x^2$
→ $3x \cdot (-12) = -36x$
✔ Answer: $-3x^3 + 6x^2 - 36x$
---
Part 6: Multiply using distributive property (FOIL for binomials)
33.) $(x - 3)(x + 4)$
→ FOIL:
First: $x \cdot x = x^2$
Outer: $x \cdot 4 = 4x$
Inner: $-3 \cdot x = -3x$
Last: $-3 \cdot 4 = -12$
→ $x^2 + 4x - 3x - 12 = x^2 + x - 12$
✔ Answer: $x^2 + x - 12$
34.) $(2x + 4)(2x + 3)$
→ First: $2x \cdot 2x = 4x^2$
→ Outer: $2x \cdot 3 = 6x$
→ Inner: $4 \cdot 2x = 8x$
→ Last: $4 \cdot 3 = 12$
→ $4x^2 + 6x + 8x + 12 = 4x^2 + 14x + 12$
✔ Answer: $4x^2 + 14x + 12$
35.) $(x - 7)(x - 6)$
→ $x \cdot x = x^2$
→ $x \cdot (-6) = -6x$
→ $-7 \cdot x = -7x$
→ $-7 \cdot (-6) = 42$
→ $x^2 - 6x - 7x + 42 = x^2 - 13x + 42$
✔ Answer: $x^2 - 13x + 42$
36.) $(3x - 1)(x + 5)$
→ $3x \cdot x = 3x^2$
→ $3x \cdot 5 = 15x$
→ $-1 \cdot x = -x$
→ $-1 \cdot 5 = -5$
→ $3x^2 + 15x - x - 5 = 3x^2 + 14x - 5$
✔ Answer: $3x^2 + 14x - 5$
37.) $(x + 5)(x^2 - 6x + 3)$
→ Distribute each term in first binomial:
→ $x(x^2 - 6x + 3) = x^3 - 6x^2 + 3x$
→ $5(x^2 - 6x + 3) = 5x^2 - 30x + 15$
→ Add:
→ $x^3 - 6x^2 + 3x + 5x^2 - 30x + 15$
→ $x^3 + (-6x^2 + 5x^2) + (3x - 30x) + 15 = x^3 - x^2 - 27x + 15$
✔ Answer: $x^3 - x^2 - 27x + 15$
38.) $(2x - 3)(4x^2 + 8x - 2)$
→ $2x(4x^2 + 8x - 2) = 8x^3 + 16x^2 - 4x$
→ $-3(4x^2 + 8x - 2) = -12x^2 - 24x + 6$
→ Add:
→ $8x^3 + 16x^2 - 4x - 12x^2 - 24x + 6$
→ $8x^3 + (16x^2 - 12x^2) + (-4x - 24x) + 6 = 8x^3 + 4x^2 - 28x + 6$
✔ Answer: $8x^3 + 4x^2 - 28x + 6$
39.) $(2x - 1)^2 = (2x - 1)(2x - 1)$
→ $2x \cdot 2x = 4x^2$
→ $2x \cdot (-1) = -2x$
→ $-1 \cdot 2x = -2x$
→ $-1 \cdot (-1) = 1$
→ $4x^2 - 2x - 2x + 1 = 4x^2 - 4x + 1$
✔ Answer: $4x^2 - 4x + 1$
40.) $(4x + 3)^2 = (4x + 3)(4x + 3)$
→ $4x \cdot 4x = 16x^2$
→ $4x \cdot 3 = 12x$
→ $3 \cdot 4x = 12x$
→ $3 \cdot 3 = 9$
→ $16x^2 + 12x + 12x + 9 = 16x^2 + 24x + 9$
✔ Answer: $16x^2 + 24x + 9$
---
✔ Final Answers Summary:
#### Part 1:
1. B
2. B
3. P
4. C
5. T
6. T
#### Part 2:
7. 0
8. 15
9. 71
10. $\frac{5}{7}$
11. $\frac{13}{10}$ or $1\frac{3}{10}$
12. $\frac{4}{15}$
#### Part 3:
12. $26x^2 + 22x + 25$
13. $-15x^2 - 21x - 11$
14. $x^2 + 32x + 18$
15. $9x^6 + 6x^5 - 15x^4 + 14$
#### Part 4:
18. $-3x + 9$
20. $14x^2 - 3x + 9$
21. $23x^2 + 12x + 4$
22. $-33x^2 - 15$
#### Part 5:
29. $6x^2 + 12x + 42$
30. $-4x^2 + 4x$
30. $-x^3 - 5x^2$
31. $12x^5 - 15x^3 + 30x^2$
32. $-3x^3 + 6x^2 - 36x$
#### Part 6:
33. $x^2 + x - 12$
34. $4x^2 + 14x + 12$
35. $x^2 - 13x + 42$
36. $3x^2 + 14x - 5$
37. $x^3 - x^2 - 27x + 15$
38. $8x^3 + 4x^2 - 28x + 6$
39. $4x^2 - 4x + 1$
40. $16x^2 + 24x + 9$
Let me know if you'd like this formatted as a printable answer key!
Parent Tip: Review the logic above to help your child master the concept of monomial binomial trinomial polynomial worksheet.
