A polynomial is an algebraic expression made up of terms which consist of variables raised to whole number exponents and their coefficients. Example: 4x² + 3x - 5.
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What is the degree of a polynomial?
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The degree of a polynomial is the highest power of the variable in the polynomial. For example, in 5x³ + 2x² - x + 1, the degree is 3.
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What is a linear polynomial?
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A linear polynomial is of degree 1 and can be expressed in the form ax + b, where a ≠ 0. Example: 2x - 3.
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What is a quadratic polynomial?
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A quadratic polynomial is of degree 2 and is expressed as ax² + bx + c, where a ≠ 0. Example: 2x² - 3x + 4.
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What is a cubic polynomial?
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A cubic polynomial is of degree 3 and is expressed in the form ax³ + bx² + cx + d, where a ≠ 0. Example: 3x³ - 2x² + x.
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Which expressions are not polynomials?
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Expressions involving negative or fractional exponents, division by a variable, or roots of variables are not polynomials. Example: 1/x or 2 + 3√x.
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What is a zero of a polynomial?
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A zero of a polynomial p(x) is a value of x, say k, for which p(k) = 0. Example: In p(x) = x² - 4, the zeros are x = 2 and x = -2.
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How do you find the value of a polynomial?
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To find the value of a polynomial p(x) at x = k, replace x with k in the expression. Example: For p(x) = x² + 1, p(2) = 2² + 1 = 5.
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How do you factor a quadratic polynomial?
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To factor ax² + bx + c, find two numbers that multiply to ac and add to b. Example: x² - 5x + 6 factors to (x - 2)(x - 3).
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What is the standard form of a quadratic polynomial?
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The standard form is given by ax² + bx + c where a ≠ 0. Example: 3x² - 4x + 1 is in standard form.
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What is polynomial long division?
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Polynomial long division is a method used to divide polynomials similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor.
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What is synthetic division?
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Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form x - k. It is simpler than long division.
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What does the graph of a polynomial represent?
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The graph of a polynomial represents the relationship between the variable and its output, showing the zeros, turning points, and end behavior.
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What is the relation between zeros and coefficients of a quadratic polynomial?
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For a quadratic ax² + bx + c, if the roots are p and q, then p + q = -b/a and pq = c/a.
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What is a common mistake in identifying the degree of a polynomial?
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A common mistake is confusing the coefficients or lower powers as the degree. Always identify the highest power of the variable.
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How can you identify a linear polynomial?
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A polynomial is linear if it has only one variable raised to the first power and no variable is multiplied together. Example: 4x + 3.
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Where are polynomials commonly used?
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Polynomials are used in calculations related to areas, volumes, and in real-world problems involving rates of change.
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How can you find the roots of a quadratic equation?
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Roots can be found using factoring, completing the square, or the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).
