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What is the meaning of the term 'polynomial'?
What is the meaning of the term 'polynomial'?
A polynomial is an expression that is composed of variables, constants, and exponents, that are combined using mathematical operations.
Match the following properties of exponents with their definitions.
Match the following properties of exponents with their definitions.
Product property of exponents = To multiply exponential terms with the same base, keep the common base and add the exponents. Power property of exponents = To raise an exponential expression to a power, keep the same base and multiply the exponents. Product to a power property = The power property can easily be extended to include more than one factor within the parentheses. Quotient to a power property = We can also raise a quotient of exponential terms to a power. Quotient property of exponents = To divide two exponential expressions with the same base, keep the common base and subtract the exponent of the denominator from the exponent of the numerator. Zero exponent property = Any base raised to the exponent of zero is equal to 1. Property of negative exponents = If the exponent of the denominator is greater than the exponent in the numerator, the quotient property yields a negative exponent.
Adding polynomials simply involves the use of the commutative and associative properties only.
Adding polynomials simply involves the use of the commutative and associative properties only.
True
A binomial is a term using only whole number exponents on variables, with no variables in the denominator.
A binomial is a term using only whole number exponents on variables, with no variables in the denominator.
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The FOIL method stands for 'First, Outer, Inner, Last,' and is used to divide two binomials.
The FOIL method stands for 'First, Outer, Inner, Last,' and is used to divide two binomials.
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The square of a binomial (a+b)² will be a²+2ab-b² if expanded.
The square of a binomial (a+b)² will be a²+2ab-b² if expanded.
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What is the sum of (14y² + 6y – 4) + (3y² + 8y + 5)?
What is the sum of (14y² + 6y – 4) + (3y² + 8y + 5)?
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What is the result of subtracting (7x² - 14) from (8x² + 3x - 19)?
What is the result of subtracting (7x² - 14) from (8x² + 3x - 19)?
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What is the product of 4x² and (2x² - 3x + 5)?
What is the product of 4x² and (2x² - 3x + 5)?
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What is the expansion of (x+8)(x+9)?
What is the expansion of (x+8)(x+9)?
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What is the product of (4x+3) and (2x-5) using the FOIL method?
What is the product of (4x+3) and (2x-5) using the FOIL method?
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Factor the binomial 4x² + 28x.
Factor the binomial 4x² + 28x.
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Factor the binomial 8x⁴ - 40x².
Factor the binomial 8x⁴ - 40x².
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Factor the expression x³ + 2x² - 5x - 10 by grouping.
Factor the expression x³ + 2x² - 5x - 10 by grouping.
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Factor the trinomial x² - 9x + 20.
Factor the trinomial x² - 9x + 20.
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A rational expression is in simplest form or lowest terms when the numerator and denominator have no common factors (other than 1).
A rational expression is in simplest form or lowest terms when the numerator and denominator have no common factors (other than 1).
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A rational number is one that can be written as the quotient of two integers. Similarly, a rational expression is one that can be written as the quotient of two polynomials.
A rational number is one that can be written as the quotient of two integers. Similarly, a rational expression is one that can be written as the quotient of two polynomials.
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The quotient of two rational expressions is computed in the same way.
The quotient of two rational expressions is computed in the same way.
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In many fields of study, formulas and literal equations involve rational expressions, and we often need to rewrite them for various reasons. Why?
In many fields of study, formulas and literal equations involve rational expressions, and we often need to rewrite them for various reasons. Why?
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Recall that the addition and subtraction of fractions requires finding the lowest common denominator (LCD) and building equivalent fractions. What operation involves this?
Recall that the addition and subtraction of fractions requires finding the lowest common denominator (LCD) and building equivalent fractions. What operation involves this?
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Operations on rational expressions use the factoring skills reviewed earlier, along with much of what we know about rational numbers. Which operation incorporates these concepts?
Operations on rational expressions use the factoring skills reviewed earlier, along with much of what we know about rational numbers. Which operation incorporates these concepts?
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The expression as a whole is called the major fraction, and any fraction occurring in a numerator or denominator is referred to as a minor fraction. What is this process called?
The expression as a whole is called the major fraction, and any fraction occurring in a numerator or denominator is referred to as a minor fraction. What is this process called?
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What are the three steps of multiplication and division of rational expressions?
What are the three steps of multiplication and division of rational expressions?
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Square roots and cube roots come from a much larger family called radical expressions. Expressions containing radicals can be found in virtually every field of mathematical study, and are an invaluable tool for modeling many real world phenomena. Why are they so important?
Square roots and cube roots come from a much larger family called radical expressions. Expressions containing radicals can be found in virtually every field of mathematical study, and are an invaluable tool for modeling many real world phenomena. Why are they so important?
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In section r.1, the square root of a is b only if b² = a. All numbers greater than zero have two square roots, one positive and one negative. The positive root is also called the principal square root. What is this concept related to?
In section r.1, the square root of a is b only if b² = a. All numbers greater than zero have two square roots, one positive and one negative. The positive root is also called the principal square root. What is this concept related to?
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As an alternative to radical notation, a rational (fractional) exponent is often used, along with the power property of exponents. What does this alternative representation enable?
As an alternative to radical notation, a rational (fractional) exponent is often used, along with the power property of exponents. What does this alternative representation enable?
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The properties used to simplify radical expressions are closely connected to the properties of exponents. Why is this connection important?
The properties used to simplify radical expressions are closely connected to the properties of exponents. Why is this connection important?
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The multiplication of radical expressions is simply an extension of our earlier work with the product property of radicals. When is this multiplication particularly useful?
The multiplication of radical expressions is simply an extension of our earlier work with the product property of radicals. When is this multiplication particularly useful?
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A right triangle is one that has a right angle. The longest side (opposite the right angle) is called the hypotenuse while the other two sides are simply called "legs." What does this relate to?
A right triangle is one that has a right angle. The longest side (opposite the right angle) is called the hypotenuse while the other two sides are simply called "legs." What does this relate to?
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Can also be established using exponential properties, in much the same way as the product property. What concept is this referring to?
Can also be established using exponential properties, in much the same way as the product property. What concept is this referring to?
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Refers to combining expressions that contain square roots (or other roots). To do this, you must ensure the radicands (the numbers under the radicals) are the same, similar to combining like terms. What is this process called?
Refers to combining expressions that contain square roots (or other roots). To do this, you must ensure the radicands (the numbers under the radicals) are the same, similar to combining like terms. What is this process called?
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A mathematical statement that expresses the equality between two linear expressions.
A mathematical statement that expresses the equality between two linear expressions.
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Is an expression that shows the relationship between two expressions using inequality signs (>, <, ≥, or ≤).
Is an expression that shows the relationship between two expressions using inequality signs (>, <, ≥, or ≤).
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It is another common form of a linear equation.
It is another common form of a linear equation.
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The property says, “If the product of any two (or more) factors is equal to zero, then at least one of the factors must be equal to zero.”
The property says, “If the product of any two (or more) factors is equal to zero, then at least one of the factors must be equal to zero.”
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It is the sum of a real number and an imaginary number.
It is the sum of a real number and an imaginary number.
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Study Notes
Polynomials and Exponents
- Polynomials are expressions composed of variables, constants, and exponents, combined using mathematical operations
- A polynomial is defined as an expression composed of variables, constants, and exponents, combined using mathematical operations.
- The power property: Raising an exponential expression to a power keeps the same base and multiplies the exponents.
- The product property: Multiplying exponential terms with the same base results in keeping the same base and adding the exponents.
- The quotient property: Dividing exponential expressions with the same base involves keeping the same base and subtracting the exponent of the denominator from the numerator
- If an exponent of a denominator is greater than the exponent in the numerator, the quotient property yields a negative exponent.
- Any base raised to the power of zero equals 1 (zero exponent property)
- Polynomial addition involves using the commutative and associative properties
- A binomial is a term using only whole number exponents on variables, with no variables in the denominator
- FOIL method is used for multiplying two binomials (First, Outer, Inner, Last)
- The square of a binomial (a+b)² when expanded equals a² + 2ab + b².
Operations on Polynomials
- Combining like terms in polynomials involves addition or subtraction of terms with identical variables and exponents
- Multiplication of a polynomial by a monomial involves applying the distributive property
- Multiplying a polynomial by a binomial involves distributing each term of one polynomial to every term of the other.
- Solving polynomial equations using the FOIL method involves multiplying the First, Outer, Inner, and Last terms of two binomials before combining like terms.
- Factoring polynomials involves finding the greatest common factor (GCF) and expressing the polynomial as a product of its factors
- Factoring binomials involves expressing the binomial as a product of its factors
Rational Expressions
- A rational expression is in simplest form when its numerator and denominator have no common factors (other than 1).
- Multiplication and division of rational expressions use factoring and reducing common factors.
- Rational number can be written as a quotient of two integers. A rational expression can be written as a quotient of two polynomials.
Radical Expressions
- Radical expressions involve square roots, cube roots, and other roots
- Simplify radical expressions by applying the multiplication and division properties of radicals.
- Radical expressions are used to model various mathematical phenomena. The properties of radicals are closely connected to the properties of exponents.
- The multiplication of radical expressions extends the product property of radicals.
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Description
Test your understanding of polynomials and exponents with this quiz. Explore properties like the power, product, and quotient along with polynomial addition techniques. Ideal for students learning algebra concepts in class.
