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Questions and Answers
What is a polynomial?
What is a polynomial?
- A mathematical expression that can include variables raised to fractional powers. (correct)
- A mathematical expression that contains only variables and no constants.
- A mathematical expression involving only constants and variables with exponents greater than one.
- A mathematical expression that consists of terms with non-negative integer exponents.
Which of the following is a polynomial?
Which of the following is a polynomial?
- 3x^2 + 2x - 1 (correct)
- x^(-3) + 7
- 4/x + 2
- √x + 5
What determines the degree of a polynomial?
What determines the degree of a polynomial?
- The highest exponent of any term in the polynomial. (correct)
- The sum of the coefficients of all terms in the polynomial.
- The lowest exponent of any term in the polynomial.
- The number of terms present in the polynomial.
Which of these expressions is not a polynomial?
Which of these expressions is not a polynomial?
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How can you combine like terms in a polynomial?
How can you combine like terms in a polynomial?
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Study Notes
Definition of a Polynomial
- A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents.
- The general form is ( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 ), where ( a ) represents coefficients, and ( n ) is a non-negative integer.
Identification of Polynomials
- Expressions with whole number exponents and no variables in the denominator qualify as polynomials.
- Examples of polynomials include ( 3x^2 + 2x + 1 ) and ( 5y^3 - 4y + 7 ).
- Non-examples include expressions like ( \frac{1}{x} ) or ( x^{-2} ) since these contain negative or fractional exponents.
Degree of a Polynomial
- The degree of a polynomial is determined by the highest power of the variable in the expression.
- For instance, in ( 4x^3 + 2x^2 + x ), the degree is 3 since ( x^3 ) is the highest exponent.
Non-Polynomial Expressions
- An expression containing variables under a radical, in the denominator, or with negative exponents is not classified as a polynomial.
- For example, ( \sqrt{x} + 2x^2 ) and ( x^{-1} + 3 ) are not polynomials.
Combining Like Terms
- Like terms in a polynomial share the same variable raised to the same power.
- To combine them, sum their coefficients while keeping the variable component unchanged, e.g., ( 3x^2 + 2x^2 = 5x^2 ).
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