Chapter 5 - Polynomials and Polynomial Functions

5.3 Factoring

100

Factor: x²-16x+60

(x-6)(x-10)

200

Factor: x³+5x²-4x-20

(x+5)(x+2)(x-2)

300

Factor: 6x⁴-18x³+12x²

(6x²)(x-2)(x-1)

400

Factor: 27x³-64

(3x-4)(9x²+12x+16)

500

Factor: x⁵-2x⁴-4x³+8x²

(x²)(x-2)²(x+2)

5.4 Long Division

100

Divide using long division: (x²+x-20)/(x+5)

x-4

200

Divide using long division: (9x³+6x²-18x+3)/(3x+2)

Quotient: 3x²-6

Remainder: 15

300

Divide using long division: (12x⁴+7x³-58x²-28x+40)/(3x-2)

4x³+5x²-16x-20

400

Use long division to find if (x²+3) is a factor of x⁴+2x²-x-4.

No, (x²+3) is not a factor of x⁴+2x²-x-4.

500

Divide using long division: (4x³-6x²-12x-8)/(2x²+x+6)

Quotient: 2x-4

Remainder: -20x + 16

5.4 Synthetic Division

100

Divide using synthetic division: (x²-9x+20)/(x-4)

x-5

200

Divide using synthetic division: (2x³-7x²-36x-60)/(x-7)

Quotient: 2x²+7x+13

Remainder: 31

300

Use synthetic division to find the value of P(x) = -3x⁴+24x³-24x²-6x+12 for x = 6.

P(6) = 408

400

Solve all roots of 2x³-3x²-18x-8 if (x+2) is a linear factor. Use synthetic division.

x = -2, -1/2, 4

500

Solve for ALL roots of x⁵+4x⁴-26x³-86x²-27x-90. (Hint: use a graphing calculator to find the real roots and synthetically divide those roots)

x = -6, -3, 5, i, -i

5.5-5.6 Theorems and Rules

100

Use the Fundamental Theorem of Algebra to find the number of total roots of x²-2x+8x⁴+12-x⁵.

5 total roots

200

List all possible rational roots of 2x⁶+4x⁴-6x³-18x²+7x+9.

1, -1, 3, -3, 9, -9, 1/2, -1/2, 3/2, -3/2, 9/2, -9/2

300

Find the number of positive and negative real roots of x⁵+3x⁴-6x²+2x-12 using Descartes' Rule of Signs.

# of positive real roots: 3 or 1

# of negative real roots: 2 or 0

400

Using the Conjugate Root Theorem, find the equation with the roots 5, sqrt(3), and -4i.

x⁵-5x⁴+13x³-65x²-48x+240

500

Solve for all roots of x⁶+x⁵-10x⁴-4x³+19x²-5x+30. (Hint: Use the Rational Root Theorem and synthetically divide the rational roots to make it easier to simplify after)

x = 2, -3, sqrt(5), -sqrt(5), i, -i

5.7 Expanding

100

Find the degrees of a and b in the expanded form of (a+b)⁷. Use P₀, P₁, P₂... in place of the coefficients

P₀a⁷+P₁a⁶b+P₂a⁵b²+P₃a⁴b³+P₄a³b⁴+P₅a²b⁵+P₆ab⁶+P₇b⁷

200

Find the coefficients of the expanded form of a binomial raised to the 8th power.

1  8  28  56  70  56  28  8  1

300

Expand: (a+b)⁵

a⁵+5a⁴b+10a³b²+10a²b³+5ab⁴+b⁵

400

Expand: (3x-y)⁴

81x⁴-108x³y+54x²y²-12xy³+y⁴

500

Expand: (2a+3b)⁶

64a⁶+576a⁵b+2160a⁴b²+4320a³b³+4860a²b⁴+2916ab⁵+729b⁶