Unit 4, Lesson 3

relative maximum

the greatest y-value within the domain

multiplicity

the number of times a multiple zero occurs

depressed polynomial

the quotient when a polynomial is divided by one of its binomial factors x - r; has a degree less than the original polynomial

How is a remainder dealt with when dividing polynomials by binomials?

the remainder is placed over the divisor to create a fraction, which is then added to the quotient

In some cases, the division of polynomials

won't come out evenly

How to Use Synthetic Division

1. Bring down the first number to begin the quotient 2. Multiply the constant in the divisor by the first number in the quotient 3. Place the result under the second coefficient and then add 4. Multiply the result by the constant in the divisor 5. Write the result under the third coefficient and then add

How to Factor a Polynomial with an Exponent Greater than 2

1. Factor the GCF out of the function 2. Factor the remaining quadratic to find the zeros

How to Write a Polynomial in Standard Form from the Product of Three Linear Factors

1. Multiply the 2nd & 3rd binomials 2. Distribute both terms in the binomial & multiply 3. Simplify the expression by combining like terms

How to Divide a Polynomial by a Binomial Using Long Division

1. Reorder the divisor & dividend into standard form, then set up the division 2. Ignore all of the terms except for the first term in the divisor and the dividend 3. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient 4. Take the first term in the quotient and multiply it by the entire divisor and then subtract 5. Carry down the last term from the dividend 6. Repeat the process by dividing the first term in the dividend by the first term in the bottom line of the division 7. Multiply the second term in the quotient by the entire divisor & subtract

How to Write a Function With Only the Zeros

1. Write linear factors for each zero 2. Multiply the second and third linear terms 3. Distribute the first linear term 4. Simplify

Factor Theorem

A special case of the Remainder Theorem The binomial x - r is a factor of the polynomial P(x) if and only if P(r) = 0

Synthetic Division

Can be used if you're dividing by a linear factor Process omits all the variables and exponents The sign of the constant term in the divisor is reversed The result is in a different format that can be converted into a polynomial The final number in the synthetic division is the remainder, so the solution can be expressed with the remainder over the divisor If there's a remainder, that means the divisor is not a factor of the dividend and the x-value is not a zero of the initial polynomial

Zero Product Property

Can be used to find the zeros of a polynomial in factored form Set each linear term equal to zero and solve

Remainder Theorem

If a polynomial P(x) is divided by x - r, the remainder is a constant P(r), and P(x) = (x - r) times Q(x) + P(r), where Q(x) is a polynomial with degree one less than the degree of P(x)

multiple zero

a zero that's repeated when a linear factor of a polynomial is repeated

A polynomial is in factored form once it has been

completely factored into its linear factors

The x-intercepts are called the zeros because

each value will result in the function being equal to zero

Polynomials can be expressed as a product of

linear factors, which are similar to prime numbers in that they cannot be factored any further

What is used to divide a polynomial by a binomial?

long division