Questions and Answers
What is the result of the expression (5x^2+6)-(3x^2-8)?
2x+14
What is the result of the expression (6x^5-3x^64+x+1)-(8x^5+3x^4-1)?
-2x^5-6x^4+x+2
What is the result of the expression (5x^4+6x^3-9x^2)-(-6x^4-6x^3+8x)?
11x^4+12x^3
What is the result of the expression (x^5y^2-x^3y^2+xy-1)-(x^5y^2-x^4y^2-x^3y^2-x^2y+xy-1)?
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What is the result of 2^4*2^3?
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What is the result of 7^5/7^2?
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What is the result of (-2)^4(-2)^2?
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What is the result of x^5*x^3 when x=2?
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What is the result of (-4m^2n^5)^3/(3)?
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Express 7% as a decimal.
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What are the absolute values of |x|=5?
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What are the solutions of -9+3|y|=24?
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Is -6 a solution of a > -4?
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Is 0 a solution of a > -4?
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If a car uses 32L of gas to travel 450 km, how many liters would be required to drive 800 km (to the nearest tenth)?
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What is the result of 3x(-x+5)?
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What is the result of 4z^2(3x+6)?
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What is the result of -6m^2(m^2+x)?
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What is the result of -7h^4(k^6-k^4-k^3+k)?
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What is the result of 2a(-5a^8b+a^2-12ab)?
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What is the result of (x+1)(x^2+3)?
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What is the result of (x-1/4)(x+1/4)?
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What is the result of 6(z-5)1.8?
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Solve the inequality 3x/5 > -15.
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Solve the inequality 11-x > 5 + 2x/5.
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Study Notes
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.
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