division and the are crucial tools for understanding polynomial functions. They allow us to evaluate polynomials efficiently and find their , which are key to graphing and analyzing these functions.
The and help us find polynomial zeros, both real and complex. These concepts, along with linear factorization, enable us to break down polynomials and solve real-world problems involving polynomial equations.
Polynomial Division and the Remainder Theorem
Remainder theorem for polynomial evaluation
States remainder when polynomial P(x) divided by (x−c) equals P(c)
Find remainder of P(x) divided by (x−c) by evaluating P(c)
Polynomial long division divides polynomial by linear factor (x−c)
Yields polynomial quotient and remainder
Remainder degree always less than divisor degree
Zeros and Factors of Polynomial Functions
Factor theorem for polynomial zeros
States (x−c) is factor of polynomial P(x) if and only if P(c)=0
If P(c)=0, then c is zero (root) of polynomial function P(x)
Factoring polynomial function helps identify zeros
If polynomial factored into linear factors, each factor corresponds to zero of function
Rational zero theorem applications
States if polynomial equation anxn+an−1xn−1+...+a1x+a0=0 has integer coefficients, then any rational solution qp (in lowest terms) must satisfy:
p is factor of constant term a0
q is factor of leading an
Find potential rational zeros by listing all possible qp combinations and testing each using Factor Theorem
Polynomial zeros: real and complex
Polynomial function of degree n has exactly n zeros (counting multiplicity)
Some zeros may be repeated (multiple roots)
Some zeros may be complex numbers (non-real roots)
Find all zeros of polynomial function:
Use Rational Zero Theorem to find potential rational zeros
Test each potential zero using Factor Theorem
Factor out confirmed zeros to reduce polynomial's degree
Repeat steps 1-3 until all zeros found or polynomial cannot be factored further
If polynomial cannot be factored further and has degree > 2, use complex number methods (De Moivre's Theorem) to find remaining zeros
Linear factorization theorem usage
States polynomial function P(x) with zeros r1,r2,...,rn (including multiplicity) can be written as:
P(x)=a(x−r1)(x−r2)...(x−rn), where a is non-zero constant
Construct polynomial with specified zeros:
Write each zero in form (x−ri)
Multiply all linear factors together
Expand product and simplify
Adjust leading coefficient if necessary
Analyzing Polynomial Functions
Descartes' rule of signs
Estimates number of positive and negative real roots of polynomial function
Number of positive real roots either equals number of sign changes between consecutive nonzero coefficients or is less by an even number
Number of negative real roots is number of sign changes between consecutive nonzero coefficients of P(−x) or is less by an even number
Provides upper bound for number of positive and negative real roots
Actual number of positive or negative real roots may be less than estimated
Real-world polynomial problem solving
Polynomial functions model various real-world situations:
Identify relevant variables and constants in problem
Construct polynomial equation representing given situation
Solve polynomial equation using appropriate methods (factoring, Rational Zero Theorem)
Interpret solution(s) in context of original problem, considering limitations or constraints
Characteristics of Polynomial Functions
Polynomial function structure
A polynomial function is an expression of the form P(x)=anxn+an−1xn−1+...+a1x+a0
The degree of the polynomial is the highest power of x (n in this case)
The leading coefficient is the coefficient of the highest degree term (an)
Multiplicity and behavior of zeros
Multiplicity refers to how many times a factor appears in the factored form of a polynomial
Affects the behavior of the graph near the zero
Higher multiplicity results in the graph touching but not crossing the x-axis at that zero
Key Terms to Review (10)
Coefficient: A coefficient is a numerical or constant factor that multiplies a variable in an algebraic expression. For example, in the term $3x^2$, 3 is the coefficient.
Complex Conjugate Theorem: The Complex Conjugate Theorem states that if a polynomial has real coefficients and a complex number as a root, then its complex conjugate is also a root. This implies that non-real roots of polynomials with real coefficients always occur in conjugate pairs.
Descartes’ Rule of Signs: Descartes’ Rule of Signs provides a method to determine the number of positive and negative real zeros in a polynomial function. It examines the number of sign changes in the sequence of coefficients.
Division Algorithm: The Division Algorithm states that for any polynomials $f(x)$ and $g(x) \neq 0$, there exist unique polynomials $q(x)$ (quotient) and $r(x)$ (remainder) such that $f(x) = g(x)q(x) + r(x)$, where the degree of $r(x)$ is less than the degree of $g(x)$.
Factor Theorem: The Factor Theorem states that a polynomial $f(x)$ has a factor $(x - c)$ if and only if $f(c) = 0$. It is used to determine whether a given binomial is a factor of the polynomial.
Polynomial: A polynomial is an algebraic expression composed of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. Examples include $3x^2 + 2x - 1$ and $5y^3 - y + 4$.
Rational Zero Theorem: The Rational Zero Theorem provides a method to identify possible rational zeros of a polynomial function. It states that any rational zero of the polynomial function with integer coefficients is a ratio of factors of the constant term to factors of the leading coefficient.
Remainder Theorem: The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear divisor $(x - c)$ is $f(c)$. This theorem helps to quickly determine whether $c$ is a root of the polynomial.
Synthetic division: Synthetic division is a simplified method of dividing a polynomial by a binomial of the form $(x - c)$. It uses only the coefficients of the polynomials and reduces the computational complexity compared to long division.
Zeros: Zeros of a function are the values of the variable that make the function equal to zero. For polynomial functions, they are also known as roots or solutions.