Podcast Beta
Questions and Answers
Which of the following numbers is an irrational number?
The decimal expansion of 1/3 is a non-repeating decimal.
False
What is the product of powers law of exponents?
a^m * a^n = a^(m+n)
The decimal expansion of an irrational number is always ______________.
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Match the following numbers with their classification:
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What is the degree of a cubic polynomial?
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A polynomial can be factorized only if it can be expressed as a product of binomials.
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What is the Remainder Theorem in polynomial division?
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The set of all CH polynomials forms a ______________, with operations such as addition and scalar multiplication.
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Match the following polynomial features with their descriptions:
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The Factor Theorem states that if (x - a) is a factor of a polynomial f(x), then f(a) ≠0.
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What is the formula to find the value of a in a quadratic polynomial ax^2 + bx + c?
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What is the general form of a cubic polynomial?
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Study Notes
Real Numbers
- Include all rational and irrational numbers
- Can be represented on the number line
- Examples: 1, 2, 3, π, e, √2
Irrational Numbers
- Cannot be expressed as a finite decimal or fraction
- Have an infinite number of digits that never repeat
- Examples: π, e, √2
Decimal Expansions
- A way to represent real numbers in base 10
- Can be terminating (e.g. 1/2 = 0.5) or non-terminating (e.g. 1/3 = 0.333...)
- Non-terminating decimals can be repeating (e.g. 1/3) or non-repeating (e.g. π)
Rational Numbers
- Can be expressed as a finite decimal or fraction (a/b, where a and b are integers)
- Examples: 1, 2, 3, 1/2, 3/4
- Can be represented as equivalent ratios
Laws of Exponents
- Product of Powers: a^m * a^n = a^(m+n)
- Power of a Product: (a*b)^m = a^m * b^m
- Power of a Power: (a^m)^n = a^(m*n)
- Zero Exponent: a^0 = 1 (where a is a non-zero number)
- Negative Exponent: a^(-m) = 1/a^m
Real Numbers
- Include all rational and irrational numbers
- Can be represented on the number line
- Examples: 1, 2, 3, π, e, √2
Irrational Numbers
- Cannot be expressed as a finite decimal or fraction
- Have an infinite number of digits that never repeat
- Examples: π, e, √2
Decimal Expansions
- A way to represent real numbers in base 10
- Can be terminating (e.g. 1/2 = 0.5) or non-terminating (e.g. 1/3 = 0.333...)
- Non-terminating decimals can be repeating (e.g. 1/3) or non-repeating (e.g. π)
Rational Numbers
- Can be expressed as a finite decimal or fraction (a/b, where a and b are integers)
- Examples: 1, 2, 3, 1/2, 3/4
- Can be represented as equivalent ratios
Laws of Exponents
Product of Powers
- a^m * a^n = a^(m+n)
Power of a Product
- (a*b)^m = a^m * b^m
Power of a Power
- (a^m)^n = a^(m*n)
Zero Exponent
- a^0 = 1 (where a is a non-zero number)
Negative Exponent
- a^(-m) = 1/a^m
CH Polynomials: Key Concepts and Formulas
Cubic Polynomials
- A cubic polynomial has a degree of three, meaning the highest power of the variable is three.
- The general form of a cubic polynomial is: ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.
- Examples of cubic polynomials include x^3 + 2x^2 - 7x + 1 and 2x^3 - 3x^2 - x + 4.
Factorization
- Factorization is the process of expressing a polynomial as a product of simpler polynomials.
- A polynomial can be factorized if it can be expressed as a product of binomials or trinomials.
- The example x^2 + 5x + 6 can be factorized as (x + 3)(x + 2).
Remainder Theorem
- The Remainder Theorem states that when a polynomial f(x) is divided by (x - a), the remainder is f(a).
- The theorem allows for evaluating a polynomial at a given value of x without actually dividing the polynomial.
Linear Algebra
- CH polynomials can be represented as vectors in a vector space.
- The set of all CH polynomials forms a vector space, with operations such as addition and scalar multiplication.
- CH polynomials can be represented as matrices, and operations such as multiplication can be performed using matrix multiplication.
Graphing Polynomials
- The graph of a polynomial is a continuous curve that opens upward or downward.
- The x-intercepts of the graph represent the roots of the polynomial.
- The y-intercept represents the constant term of the polynomial.
Factor Theorem
- The Factor Theorem states that if (x - a) is a factor of a polynomial f(x), then f(a) = 0.
- The theorem can be used to find the roots of a polynomial by finding the values of x that make the polynomial equal to zero.
Finding the Value of a and b
- In a quadratic polynomial ax^2 + bx + c, the values of a and b can be found using the formulas:
- a = (sum of the roots) / 2
- b = product of the roots
- These formulas can be used to find the values of a and b in a CH polynomial.
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Description
This quiz covers real numbers, including rational and irrational numbers, and their decimal expansions. Learn about terminating and non-terminating decimals and how to represent them.
