Polynomial Theorems Quiz

Factor Theorem

• States that a polynomial (p(x)) has a factor ((x-c)) if and only if (p(c) = 0).
• This theorem provides a method to determine if a given factor divides a polynomial.
• If (p(c)=0), then ((x-c)) is a factor of (p(x)).
• Conversely, if ((x-c)) is a factor of (p(x)), then (p(c)=0).
• This theorem is fundamental in polynomial factorization.

Remainder Theorem

• The remainder theorem states that when a polynomial (p(x)) is divided by ((x-c)), the remainder is (p(c)).
• This theorem allows to find the remainder without performing the division.
• It provides a direct way to evaluate a polynomial at a specific value of (x).
• The remainder is a constant.

Rational Root Theorem

• The rational root theorem provides a way to determine the possible rational roots of a polynomial equation with integer coefficients.
• It states that if a polynomial (p(x)=a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0) has rational roots, then the root can be expressed as (\frac{p}{q}), where (p) is a factor of the constant term (a_0) and (q) is a factor of the leading coefficient (a_n).
• All rational roots will be in the form (\frac{p}{q}), where (p) divides the constant term and (q) divides the leading coefficient.
• It helps narrow down the search for rational roots.
• It does not guarantee there are rational roots.
• Important to note that the theorem only identifies possible rational roots; further testing is required to confirm if any of these possible roots are actual roots.

Relationships between the theorems

• The factor theorem and remainder theorem are closely related. The remainder theorem is a direct application of polynomial division, while the factor theorem specifies a particular case of division where the remainder is zero.
• The rational root theorem provides a tool to potentially find rational roots, which in turn can be used with the factor theorem to simplify the polynomial.