Explain how the Remainder Theorem can be used to find f(-6) if f(x) = x4 + 7x3 + 8x2 +

To divide x3 + 5x2 - 7x + 1 by x - 4 using synthetic division, the first step is to write .

To divide 4x3 - 8x - 2 by x + 5 using synthetic division, the first step is to write .

True or false: -1| 3 -4 2 -1 -3 7 -9 3 -7 9 -10 means 3x3 - 4x2 + 2x - 1 x + 1 = 3x2 - 7x + 9 - 10 x + 1 . ______________

The Remainder Theorem states that if the polynomial f(x) is divided by x - c, then the remainder is ______________.

(4x3 - 3x2 + 3x - 1) , (x - 1)

(5x3 - 6x2 + 3x + 11) , (x - 2)

(6x5 - 2x3 + 4x2 - 3x + 1) , (x - 2)

(x5 + 4x4 - 3x2 + 2x + 3) , (x - 3)

(x2 - 5x - 5x3 + x4 ) , (5 + x)

(x2 - 6x - 6x3 + x4 ) , (6 + x)

(3x3 + 2x2 - 4x + 1) , ax - 1 3 b

(2x4 - x3 + 2x2 - 3x + 1) , ax - 1 2 b

x5 + x3 - 2 x - 1

x7 + x5 - 10x3 + 12 x + 2

x4 - 256 x - 4

x7 - 128 x - 2

2x5 - 3x4 + x3 - x2 + 2x - 1 x + 2

x5 - 2x4 - x3 + 3x2 - x + 1 x - 2

f(x) = 2x3 - 11x2 + 7x - 5; f(4)

f(x) = x3 - 7x2 + 5x - 6; f(3)

f(x) = 3x3 - 7x2 - 2x + 5; f(-3)

f(x) = 4x3 + 5x2 - 6x - 4; f(-2)

f(x) = x4 + 5x3 + 5x2 - 5x - 6; f(3)

f(x) = x4 - 5x3 + 5x2 + 5x - 6; f(2)

f(x) = 2x4 - 5x3 - x2 + 3x + 2; f a- 1 2 b

f(x) = 6x4 + 10x3 + 5x2 + x + 1; f a- 2 3 b

x3 - 4x2 + x + 6 = 0; -1

x3 - 2x2 - x + 2 = 0; -1

2x3 - 5x2 + x + 2 = 0; 2

2x3 - 3x2 - 11x + 6 = 0; -2

6x3 + 25x2 - 24x + 5 = 0; -5

3x3 + 7x2 - 22x - 8 = 0; -4

x3 + 2x2 - 5x - 6 = 0

2x3 + x2 - 13x + 6 = 0

6x3 - 11x2 + 6x - 1 = 0

2x3 + 11x2 - 7x - 6 = 0

(22x - 24 + 7x3 - 29x2 + 4x4 )(x + 4)-1

(9 - x2 + 6x + 2x3 )(x + 1)-1

In Exercises 3940, write a polynomial that represents the length of each rectangle.

In Exercises 3940, write a polynomial that represents the length of each rectangle.

a. Use synthetic division to show that 3 is a solution of the polynomial equation 14x3 - 17x2 - 16x - 177 = 0. b. Use the solution from part (a) to solve this problem. The number of eggs, f(x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by f(x) = 14x3 - 17x2 - 16x + 34. What is the abdominal width when there are 211 eggs?

a. Use synthetic division to show that 2 is a solution of the polynomial equation 2h3 + 14h2 - 72 = 0. b. Use the solution from part (a) to solve this problem. The width of a rectangular box is twice the height and the length is 7 inches more than the height. If the volume is 72 cubic inches, find the dimensions of the box.

Explain how to perform synthetic division. Use the division problem (2x3 - 3x2 - 11x + 7) , (x - 3) to support your explanation.

State the Remainder Theorem.

Explain how the Remainder Theorem can be used to find f(-6) if f(x) = x4 + 7x3 + 8x2 + 11x + 5. What advantage is there to using the Remainder Theorem in this situation rather than evaluating f(-6) directly?

Explain how the Remainder Theorem and synthetic division can be used to determine whether -4 is a solution of the following equation: 5x3 + 22x2 + x - 28 = 0.

For each equation that you solved in Exercises 2732, use a graphing utility to graph the polynomial function on the left side of the equation. Use end behavior to obtain a complete graph. Then use the graphs x@intercepts to verify your solutions.

There are certain kinds of polynomial divisions that I can perform using long division, but not using synthetic division.

Every time I divide polynomials using synthetic division, I am using a highly condensed form of the long division procedure where omitting the variables and exponents does not involve the loss of any essential data.

The only nongraphic method that I have for evaluating a function at a given value is to substitute that value into the functions equation.

The Remainder Theorem gives me a method for factoring certain polynomials that I could not factor using the factoring strategies from the previous chapter.

Synthetic division is a process for dividing a polynomial by x - c. The coefficient of x is 1. How might synthetic division be used if you are dividing by 2x - 4?

Use synthetic division to show that 5 is a solution of x4 - 4x3 - 9x2 + 16x + 20 = 0. Then solve the polynomial equation. Hint: Use factoring by grouping when working with the quotient factor.

Solve: 4x + 3 - 13x - 7 6 2(3 - 4x). (Section 4.1, Example 3)

Solve: 2x(x + 3) + 6(x - 3) = -28. (Section 5.7, Example 2)

Solve by Cramers rule: e 7x - 6y = 17 3x + y = 18. (Section 3.5, Example 2)

x + 4 2x , x + 20 3x

x 3 , 9 x , 4

2x x - 3 , 6 x + 3 , - 28 x2 - 9