Midterm Review - Ch 4 & 5

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What is polynomial long division?

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Polynomial long division is a method used to divide a polynomial by another polynomial of the same or lower degree, similar to long division with numbers. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend.

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How do you add polynomials?

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To add polynomials, combine like terms by adding their coefficients. For example, to add (3x^2 + 2x + 1) and (4x^2 + 3), you would combine the x^2 terms (3x^2 + 4x^2 = 7x^2), the x terms (2x + 0 = 2x), and the constant terms (1 + 3 = 4), resulting in 7x^2 + 2x + 4.

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What is the zero of a polynomial?

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A zero of a polynomial is a value of x that makes the polynomial equal to zero. For example, if P(x) = x^2 - 4, the zeros are x = 2 and x = -2, since P(2) = 0 and P(-2) = 0.

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What is the process to find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for x. This can involve factoring, using the quadratic formula, or synthetic division.

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What is the difference between a polynomial and a rational function?

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A polynomial is a mathematical expression consisting of variables raised to whole number powers and coefficients, while a rational function is a ratio of two polynomials. For example, P(x) = x^2 + 3x + 2 is a polynomial, while R(x) = (x^2 + 1)/(x - 1) is a rational function.

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What is the standard form of a polynomial?

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The standard form of a polynomial is written in descending order of the degree of its terms. For example, the polynomial 3x^3 + 2x^2 - x + 5 is in standard form.

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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 the polynomial 4x^5 + 3x^3 - 2, the degree is 5.

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What does it mean to simplify an expression?

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To simplify an expression means to reduce it to its simplest form by combining like terms, removing parentheses, and performing any possible arithmetic operations.

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What is the distributive property?

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The distributive property states that a(b + c) = ab + ac. It allows you to multiply a single term by each term inside a set of parentheses.

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What is a quadratic equation?

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A quadratic equation is a polynomial equation of degree 2, typically in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.