HW 2.5 Factoring Polynomials

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What does the degree of a polynomial indicate?

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The degree of a polynomial indicates the highest power of the variable in the polynomial, which determines the maximum number of roots (solutions) the polynomial can have.

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State the Fundamental Theorem of Algebra.

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The Fundamental Theorem of Algebra states that every non-constant polynomial equation of degree n has exactly n roots in the complex number system, counting multiplicities.

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The possible rational zeros are: -1/3, -7/3, -1, -7, 1/3, 7/3, 1, 7.

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (indeterminates) raised to non-negative integer powers and coefficients, combined using addition, subtraction, and multiplication.

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Define a root of a polynomial.

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A root of a polynomial is a value of the variable that makes the polynomial equal to zero.

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What is the difference between rational and irrational roots?

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Rational roots can be expressed as a fraction of two integers, while irrational roots cannot be expressed as a simple fraction and often involve square roots or other non-repeating decimals.

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Explain the concept of factoring a polynomial.

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Factoring a polynomial involves rewriting it as a product of simpler polynomials or expressions, which can help in finding its roots.

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What is the significance of complex roots in polynomials?

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Complex roots occur in conjugate pairs and indicate that the polynomial does not intersect the x-axis at those points, reflecting the behavior of the polynomial in the complex plane.