Complex Numbers and Solving Polynomials Test Review

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What is a complex number?

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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as √-1.

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What is the imaginary unit 'i'?

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The imaginary unit 'i' is defined as the square root of -1, which is used to extend the real number system to include solutions to equations that do not have real solutions.

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What is the standard form of a complex number?

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The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part.

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How do you add complex numbers?

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To add complex numbers, combine their real parts and their imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

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How do you subtract complex numbers?

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To subtract complex numbers, subtract their real parts and their imaginary parts separately. For example, (a + bi) - (c + di) = (a - c) + (b - d)i.

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What is the product of two complex numbers?

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The product of two complex numbers (a + bi)(c + di) is given by ac + adi + bci + bdi^2, which simplifies to (ac - bd) + (ad + bc)i.

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What is the conjugate of a complex number?

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The conjugate of a complex number a + bi is a - bi. It is used to simplify division of complex numbers.

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How do you divide complex numbers?

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To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, (a + bi) / (c + di) is multiplied by (c - di)/(c - di).

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What is the discriminant in a quadratic equation?

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The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac, which determines the nature of the roots of the equation.

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What does it mean if the discriminant is positive?

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If the discriminant is positive, the quadratic equation has two distinct real solutions.