Math Problem Statement

Expand the polynomial expression: a x ( 2x - 3a ) - x ( a x + 5a^2 )

Solution

The given problem asks to expand the polynomial expression:

ax(2x3a)x(ax+5a2).a x \left( 2x - 3a \right) - x \left( a x + 5a^2 \right).

Step-by-step Expansion:

  1. Expand the first term ax(2x3a)a x \left( 2x - 3a \right): ax(2x)=2ax2,ax(3a)=3a2x.a x (2x) = 2a x^2, \quad a x (-3a) = -3a^2 x. Thus: ax(2x3a)=2ax23a2x.a x \left( 2x - 3a \right) = 2a x^2 - 3a^2 x.

  2. Expand the second term x(ax+5a2)-x \left( a x + 5a^2 \right): x(ax)=ax2,x(5a2)=5a2x.-x (a x) = -a x^2, \quad -x (5a^2) = -5a^2 x. Thus: x(ax+5a2)=ax25a2x.-x \left( a x + 5a^2 \right) = -a x^2 - 5a^2 x.

  3. Combine the expanded terms: (2ax23a2x)+(ax25a2x).\left( 2a x^2 - 3a^2 x \right) + \left( -a x^2 - 5a^2 x \right).

  4. Simplify by combining like terms:

    • Combine x2x^2-terms: 2ax2ax2=ax22a x^2 - a x^2 = a x^2.
    • Combine xx-terms: 3a2x5a2x=8a2x-3a^2 x - 5a^2 x = -8a^2 x.

    Final expanded polynomial: ax28a2x.a x^2 - 8a^2 x.

Final Answer:

ax28a2x.a x^2 - 8a^2 x.

Would you like further clarification or additional steps?


Follow-up Questions:

  1. How does distributing terms work in polynomial expansion?
  2. What are some strategies to identify like terms in a polynomial?
  3. Could this expression be factored back into its original form? If so, how?
  4. What role does the distributive property play in simplifying polynomials?
  5. How would the solution change if the coefficients were functions instead of constants?

Tip:

Always double-check your work when combining terms to ensure accuracy in coefficients and variables!

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Expansion
Distributive Property
Combining Like Terms

Formulas

Distributive property: a(b + c) = ab + ac

Theorems

Distributive property

Suitable Grade Level

Grades 8-10