Math Problem Statement

a) Tính A(x) + B(x) b) Tính A(x) – B(x)

a) Tính A(x) + B(x) b) Tính A(x) - B(x)

Solution

Let's analyze the image to extract the mathematical expressions for A(x)A(x) and B(x)B(x).

From the image, we can see the following functions:

A(x)=5x4+x3+12x2+2x+2A(x) = 5x^4 + x^3 + \frac{1}{2}x^2 + 2x + 2 B(x)=5x4x3+12x210x3B(x) = 5x^4 - x^3 + \frac{1}{2}x^2 - 10x - 3

a) Tính A(x)+B(x)A(x) + B(x)

A(x)+B(x)=(5x4+x3+12x2+2x+2)+(5x4x3+12x210x3)A(x) + B(x) = (5x^4 + x^3 + \frac{1}{2}x^2 + 2x + 2) + (5x^4 - x^3 + \frac{1}{2}x^2 - 10x - 3)

Let's combine like terms:

A(x)+B(x)=(5x4+5x4)+(x3x3)+(12x2+12x2)+(2x10x)+(23)A(x) + B(x) = (5x^4 + 5x^4) + (x^3 - x^3) + \left(\frac{1}{2}x^2 + \frac{1}{2}x^2\right) + (2x - 10x) + (2 - 3)

Simplifying:

A(x)+B(x)=10x4+0x3+x28x1A(x) + B(x) = 10x^4 + 0x^3 + x^2 - 8x - 1

So:

A(x)+B(x)=10x4+x28x1A(x) + B(x) = 10x^4 + x^2 - 8x - 1

b) Tính A(x)B(x)A(x) - B(x)

A(x)B(x)=(5x4+x3+12x2+2x+2)(5x4x3+12x210x3)A(x) - B(x) = (5x^4 + x^3 + \frac{1}{2}x^2 + 2x + 2) - (5x^4 - x^3 + \frac{1}{2}x^2 - 10x - 3)

Now distribute the negative sign:

A(x)B(x)=5x4+x3+12x2+2x+25x4+x312x2+10x+3A(x) - B(x) = 5x^4 + x^3 + \frac{1}{2}x^2 + 2x + 2 - 5x^4 + x^3 - \frac{1}{2}x^2 + 10x + 3

Combine like terms:

A(x)B(x)=(5x45x4)+(x3+x3)+(12x212x2)+(2x+10x)+(2+3)A(x) - B(x) = (5x^4 - 5x^4) + (x^3 + x^3) + \left(\frac{1}{2}x^2 - \frac{1}{2}x^2\right) + (2x + 10x) + (2 + 3)

Simplifying:

A(x)B(x)=0x4+2x3+0x2+12x+5A(x) - B(x) = 0x^4 + 2x^3 + 0x^2 + 12x + 5

So:

A(x)B(x)=2x3+12x+5A(x) - B(x) = 2x^3 + 12x + 5

Final Answers:

  • A(x)+B(x)=10x4+x28x1A(x) + B(x) = 10x^4 + x^2 - 8x - 1
  • A(x)B(x)=2x3+12x+5A(x) - B(x) = 2x^3 + 12x + 5

Would you like further details or explanations?

Here are 5 related questions for practice:

  1. What is A(x)×B(x)A(x) \times B(x)?
  2. How would you calculate A(x)B(x)\frac{A(x)}{B(x)}?
  3. Can you find the roots of A(x)B(x)=0A(x) - B(x) = 0?
  4. What is the degree of the polynomial A(x)+B(x)A(x) + B(x)?
  5. How would the solution change if A(x)A(x) or B(x)B(x) included a term with x5x^5?

Tip: When simplifying polynomials, always group like terms first to avoid mistakes with signs or coefficients.

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Math Problem Analysis

Mathematical Concepts

Polynomial Addition
Polynomial Subtraction
Algebra

Formulas

A(x) + B(x) = (5x^4 + x^3 + 1/2x^2 + 2x + 2) + (5x^4 - x^3 + 1/2x^2 - 10x - 3)
A(x) - B(x) = (5x^4 + x^3 + 1/2x^2 + 2x + 2) - (5x^4 - x^3 + 1/2x^2 - 10x - 3)

Theorems

Polynomial Addition Theorem
Polynomial Subtraction Theorem

Suitable Grade Level

Grades 9-11