Hard Math Problems Flashcards

Polynomial Operations

• Simplifying polynomials involves combining like terms and applying distributive properties.
• For the expression (5x^2+6)-(3x^2-8), the result after simplification is 2x + 14.
• The expression (6x^5-3x^4+x+1)-(8x^5+3x^4-1) simplifies to -2x^5 - 6x^4 + x + 2.

Higher-Degree Polynomials

• Combining polynomials with higher degrees requires careful tracking of each term.
• The expression (5x^4+6x^3-9x^2)-(-6x^4-6x^3+8x) simplifies to 11x^4 + 12x^3.

Multivariable Expressions

• Involving variables such as (x^5y^2-x^3y^2+xy-1)-(x^5y^2-x^4y^2-x^3y^2-x^2y+xy-1) simplifies to x^4y^2 + x^2y.

Exponents and Powers

• The product of powers can be expressed as 2^4 * 2^3 = 2^7.
• Division of powers: 7^5 / 7^2 reduces to 7^3.
• Multiplying powers: (-2)^4 * (-2)^2 results in 64.

Evaluating Expressions

• For x=2 in x^5*x^3, the evaluation yields 256.
• Evaluating powers can sometimes lead to fractions or negative numbers.

Absolute Values

• The absolute value equation |x| = 5 has solutions 5 and -5.
• The equation -9 + 3|y| = 24 gives solutions 11 and -11.

Inequalities and Solutions

• Testing numbers against inequalities: -6 does not satisfy a > -4, while 0 does.

Applications in Real Life

• A car consuming 32L to travel 450 km will need 56.9L for 800 km.

Algebraic Expansion

• Simple algebraic expressions expand using the distributive property.
• For example, 3x(-x + 5) simplifies to -3x^2 + 15x.
• Additionally, 4z^2(3x + 6) expands to 12x^3 + 24x^2.

Factorization

• (x + 1)(x^2 + 3) expands into x^3 + x^2 + 3x + 3.
• Factorization also applies to smaller expressions like (x - 1/4)(x + 1/4), which yields x^2 - 1/16.

Solving Inequalities

• Solving the inequality 3x/5 > -15 leads to x > -25.
• For 11 - x > 5 + 2x/5, you find equivalent expressions or bounds for x.