Math Problem Statement

How do you solve problems involving adding and subtracting polynomials, as shown in the provided worksheet?

Solution

I see that you've uploaded a worksheet titled 10.1 Adding and Subtracting Polynomials. Let me help you solve these problems step by step. Here's a breakdown:


1. (4x² - 2x + 8) + (x² + 3x - 2)

Combine like terms: (4x2+x2)+(2x+3x)+(82)=5x2+x+6(4x^2 + x^2) + (-2x + 3x) + (8 - 2) = 5x^2 + x + 6 Answer: 5x2+x+65x^2 + x + 6


2. (4x² - 2x + 8) + (2x² + 2x - 5)

Combine like terms: (4x2+2x2)+(2x+2x)+(85)=6x2+0x+3=6x2+3(4x^2 + 2x^2) + (-2x + 2x) + (8 - 5) = 6x^2 + 0x + 3 = 6x^2 + 3 Answer: 6x2+36x^2 + 3


3. (5x² - x + 5) + (x² + x - 5)

Combine like terms: (5x2+x2)+(x+x)+(55)=6x2+0x+0=6x2(5x^2 + x^2) + (-x + x) + (5 - 5) = 6x^2 + 0x + 0 = 6x^2 Answer: 6x26x^2


4. (3x² - 3x + 4) + (x² + 3x - 6)

Combine like terms: (3x2+x2)+(3x+3x)+(46)=4x2+0x2=4x22(3x^2 + x^2) + (-3x + 3x) + (4 - 6) = 4x^2 + 0x - 2 = 4x^2 - 2 Answer: 4x224x^2 - 2


5. (2x² - 2x + 3) - (x² + 5x - 5)

Distribute the subtraction and combine like terms: (2x2x2)+(2x5x)+(3(5))=x27x+8(2x^2 - x^2) + (-2x - 5x) + (3 - (-5)) = x^2 - 7x + 8 Answer: x27x+8x^2 - 7x + 8


6. (2x² - 4x + 8) - (x² + 3x - 1)

Distribute the subtraction and combine like terms: (2x2x2)+(4x3x)+(8(1))=x27x+9(2x^2 - x^2) + (-4x - 3x) + (8 - (-1)) = x^2 - 7x + 9 Answer: x27x+9x^2 - 7x + 9


7. (6x² - 4x + 1) - (2x² + 6x - 1)

Distribute the subtraction and combine like terms: (6x22x2)+(4x6x)+(1(1))=4x210x+2(6x^2 - 2x^2) + (-4x - 6x) + (1 - (-1)) = 4x^2 - 10x + 2 Answer: 4x210x+24x^2 - 10x + 2


8. (6x² - 2x + 2) - (5x² + 2x - 2)

Distribute the subtraction and combine like terms: (6x25x2)+(2x2x)+(2(2))=x24x+4(6x^2 - 5x^2) + (-2x - 2x) + (2 - (-2)) = x^2 - 4x + 4 Answer: x24x+4x^2 - 4x + 4


9. (7x² - 5x + 11) - (3x² + 2x - 11)

Distribute the subtraction and combine like terms: (7x23x2)+(5x2x)+(11(11))=4x27x+22(7x^2 - 3x^2) + (-5x - 2x) + (11 - (-11)) = 4x^2 - 7x + 22 Answer: 4x27x+224x^2 - 7x + 22


10. (7x² - 8x + 10) - (2x² + x + 10)

Distribute the subtraction and combine like terms: (7x22x2)+(8xx)+(1010)=5x29x(7x^2 - 2x^2) + (-8x - x) + (10 - 10) = 5x^2 - 9x Answer: 5x29x5x^2 - 9x


11. State the degree of the monomial 7x47x^4.

The degree of a monomial is the highest power of the variable. Here, the degree is 4.

Answer: 44


12. Find the product of 2(x3+5)2(x^3 + 5).

Distribute the 22: 2(x3)+2(5)=2x3+102(x^3) + 2(5) = 2x^3 + 10 Answer: 2x3+102x^3 + 10


Would you like further clarification or more detailed steps for any problem?

Here are five related questions to expand your understanding:

  1. How do you determine the degree of a polynomial?
  2. What is the process for combining like terms in a subtraction problem?
  3. How can distributing negative signs affect polynomial subtraction?
  4. How do you multiply a monomial with a polynomial?
  5. Can a polynomial have terms with fractional or negative exponents?

Tip: Always align like terms carefully while adding or subtracting polynomials to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Polynomial addition
Polynomial subtraction
Combining like terms
Distributive property

Formulas

Combine like terms: ax^n + bx^n = (a + b)x^n
Distribute: a(b + c) = ab + ac

Theorems

Fundamental theorem of algebra for polynomial operations

Suitable Grade Level

Grade 8-10