Polynomials ~ Factor and Solve.

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What is a polynomial?

Back

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

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What are the degrees of polynomials?

Back

The degree of a polynomial is the highest power of the variable in the polynomial. For example, in the polynomial x^3 + 2x^2 + 3, the degree is 3.

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What is the Factor Theorem?

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The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0, meaning c is a root of the polynomial.

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How do you find the roots of a polynomial?

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

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor of the form (x - c).

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What is the difference between real and complex roots?

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Real roots are solutions to a polynomial equation that are real numbers, while complex roots include imaginary numbers and are of the form a + bi, where a and b are real numbers.

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What is the quadratic formula?

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The quadratic formula is used to find the roots of a quadratic equation ax^2 + bx + c = 0 and is given by x = (-b ± √(b² - 4ac)) / (2a).

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What is the process of factoring a polynomial?

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Factoring a polynomial involves rewriting it as a product of its factors, which can include linear factors and irreducible polynomials.

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What is the significance of the discriminant in a quadratic equation?

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The discriminant (b² - 4ac) determines the nature of the roots of a quadratic equation: if it is positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.

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What is the Rational Root Theorem?

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The Rational Root Theorem states that any rational solution (or root) of a polynomial equation with integer coefficients is of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.