Math Problem Statement

Solve the following set of 13 math problems, covering topics in algebra, polynomials, and equations.

Solution

The image contains a list of 13 mathematics problems in Polish, covering topics related to algebra, polynomials, and equations. Below is an overview of each problem:

  1. Simplify Expression: Simplify (2a5b)2+(a5b)(a+5b)2a(a+5b)(2a - 5b)^2 + (a - 5b)(a + 5b) - 2a(a + 5b).
  2. Evaluate Polynomial: Find the value of the polynomial 128x348x2(18x)128x^3 - 48x^2(1 - 8x) at x=14x = \frac{1}{4}.
  3. Polynomial Equality: Determine which polynomial is equal to (x8)(x+8)+(3x8)2(x - 8)(x + 8) + (3x - 8)^2.
  4. Evaluate Polynomial Value: Calculate W(x)=(8x6+134x512x4+2x3+x+1)W(x) = (-8x^6 + \frac{13}{4}x^5 - \frac{1}{2}x^4 + 2x^3 + x + 1) at x=15x = -\frac{1}{5}.
  5. Find Coefficients: Find coefficients aa and bb for the polynomial W(x)=x3ax2x+bW(x) = x^3 - ax^2 - x + b given W(0)=1W(0) = 1 and W(1)=3W(1) = 3.
  6. Polynomial Division: Given polynomials W(x)=3x+6W(x) = -3x + 6, P(x)=x22x+1P(x) = x^2 - 2x + 1, and Q(x)=5x3+x4Q(x) = 5x^3 + x - 4, find the form of 13W(x)(P(x)+Q(x))\frac{1}{3} W(x)(P(x) + Q(x)).
  7. Polynomial Operations: Simplify the expression (2(34x+x3)(2y)(x2+6x3))(2x3)(2(3 - 4x + x^3) - (2 - y)(x^2 + 6x - 3))(2x - 3).
  8. Determine Missing Factor: Identify which factor does not appear in the factorization of C(x)=(9x21)(x24)C(x) = (9x^2 - 1)(x^2 - 4).
  9. Factorize Polynomial: Factor W(x)=4x316x2+12xW(x) = 4x^3 - 16x^2 + 12x and determine its form.
  10. Solution Verification: Identify which value is a solution to the given complex equation.
  11. Solve Equation: Solve x2(2x5)=(9x4)(x+1)x^2(2x - 5) = (9x - 4)(x + 1).
  12. Find All Solutions: Solve 3x(5x2+1)=(3x)(5x2+4)5x(x24)3x(5x^2 + 1) = (3x)(5x^2 + 4) - 5x(x^2 - 4).
  13. Count Rational Solutions: Determine how many rational solutions satisfy x3(x7)(x210)(x2+x+1)=0x^3(x - \sqrt{7})(x^2 - 10)(x^2 + x + 1) = 0.

Would you like detailed solutions to any specific problems, or would you like me to start solving each one?

Here are five relative questions based on the concepts in this problem set:

  1. How can we factor polynomials involving higher powers or multiple terms effectively?
  2. What methods are best for simplifying expressions involving multiple variables?
  3. How can we identify rational solutions for polynomial equations?
  4. What are the typical steps to solve a polynomial equality problem?
  5. How can we determine which terms or factors are missing in polynomial factorizations?

Tip: For polynomials, the Rational Root Theorem can be a helpful tool to identify possible rational roots when factoring.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Simplification
Equation Solving
Factoring
Polynomial Division
Rational Root Theorem

Formulas

Polynomial Evaluation
Factoring Techniques
Simplification of Expressions
Quadratic Formula

Theorems

Fundamental Theorem of Algebra
Rational Root Theorem

Suitable Grade Level

Grades 9-12