Factors, Remainders, and Roots, Oh My!

1 November 2010

Remainders

Is there any way I can figure out my remainder in advance?

(3x4 – 8x3 + 9x + 5) ÷ (x – 2) 3x3 – 2x2 – 4x + 1 Remainder 7

Remainder Theorem

If a polynomial f(x) is divided by x – c, then the remainder is f(c). Like synthetic division, the divisor

must be in the form x – c. If it isn’t, we must alter to the divisor to include subtraction.

Remainder Theorem, cont.

(3x4 – 8x3 + 9x + 5) ÷ (x – 2) f(2) = 3(2)4 – 8(2)3 + 9(2) + 5 f(2) = 7

Remainder Theorem, cont.

(2x4 + 5x3 − 2x − 8) ÷ (x + 3) x + 3 x – (-3)

f(-3) = 2(-3)4 + 5(-3)3 – 2(-3) – 8 f(-3) = 25

Your Turn

On page 249 in your textbook, complete problems 10 – 16. You will Solve for the quotient using synthetic

division Check your remainder using the

Remainder Theorem

Remainders and Factors

If a polynomial f(x) is divided by x – a, and f(a) = 0, then x – a is a factor of the polynomial.

The Factor Theorem

Remainders and Factors, cont.

Similarly, if a divisor has a remainder of zero, than the quotient is also a factor of the polynomial.

Remainders and Factors, cont.

Ex. (a4 – 1) ÷ (a – 1) = a3 + a2 + a + 1

Both a – 1 and a3 + a2 + a + 1 are factors of a4 – 1!

Your Turn:

On pg. 249 in your textbook, complete problems 41 – 46. You will: Use the Factor Theorem to determine

if the given h(x) is a factor of f(x). Confirm your results using synthetic

division.

Maximum Number of Roots

A polynomial of degree n has at most n different roots.

Example: f(x) = x2 – 3x + 4 has at most 2

different roots 0 = (x – 3)(x – 1); x = 1, 3

Maximum Number of Roots, cont.

However, a polynomial can have less than the maximum number of different roots.

This is because roots can repeat. Example: f(x) = x2 – 10x + 25

0 = (x – 5)(x – 5); x = 5

Other Roots Connections

Let f(x) be a polynomial. If r is a real number for which one of the following statements is true, then all of the following statements are true: r is a zero of f(x)

Other Roots Connections, cont.

r is an x-intercept of f(x) x = r is a solution or root when f(x) =

0 x – r is a factor of the polynomial f(x)

Applications

We can use the maximum number of roots and the root connections to construct the equation of a polynomial from its graph.

Applications, cont. x-intercepts: Zeros: Solutions: Max Degree: Linear Factors:

Applications, cont. Linear Factors:

(x+1)(x – 3)

Equation:

Your Turn: On page 249 in your textbook,

complete problems 51 – 53. You will: List the x-intercepts List the zeros List the solutions Determine the maximum degree Product of the linear factors Determine the equation of a graph

Hmwk:

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