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Questions and Answers
What is the expanded form of (x + 2)^4 using Pascal's Triangle?
What is the expanded form of (x + 2)^4 using Pascal's Triangle?
The expanded form is $x^4 + 8x^3 + 24x^2 + 32x + 16$.
How can you completely factor the polynomial 2x^3 - 18x?
How can you completely factor the polynomial 2x^3 - 18x?
The factored form is $2x(x^2 - 9) = 2x(x - 3)(x + 3)$.
What is the result of cubing the polynomial (x + 3)^3?
What is the result of cubing the polynomial (x + 3)^3?
The result is $x^3 + 9x^2 + 27x + 27$.
If a phone plan costs $35 per month plus $0.10 per minute, how many minutes can you talk for $50 in a month?
If a phone plan costs $35 per month plus $0.10 per minute, how many minutes can you talk for $50 in a month?
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What is the square root of -16?
What is the square root of -16?
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What are the possible positive and negative zeros for the polynomial f(x) = x^4 - 3x^3 + 2x^2 - x + 1?
What are the possible positive and negative zeros for the polynomial f(x) = x^4 - 3x^3 + 2x^2 - x + 1?
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Identify the leading coefficient and degree of the polynomial f(x) = -2x^3 + 5x^2 - x + 7.
Identify the leading coefficient and degree of the polynomial f(x) = -2x^3 + 5x^2 - x + 7.
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Study Notes
Expanding Polynomials
- Expanding (x+2)^4 using Pascal's Triangle: Pascal's Triangle provides coefficients for binomial expansions. The fourth row of Pascal's Triangle is 1, 4, 6, 4, 1. Therefore, (x+2)^4 = 1x^4 + 4(2)x^3 + 6(2^2)x^2 + 4(2^3)x + 1(2^4) = x^4 + 8x^3 + 24x^2 + 32x + 16
Factoring Polynomials
- Factoring 2x^3 - 18x: Factor out the greatest common factor (GCF), which is 2x. This yields 2x(x^2 - 9). Then factor the difference of squares (x^2 - 9) to get 2x(x-3)(x+3)
Cubing Polynomials
- Cubing (x+3)^3: Use the binomial expansion formula (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3. Applying this to (x+3)^3, we get x^3 + 9x^2 + 27x + 27
Adding Polynomials
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Adding 2x^2 + 3x - 1 and x^2 - 2x + 5: Add like terms vertically:
2x^2 + 3x - 1
- x^2 - 2x + 5
3x^2 + x + 4
Phone Plan Calculation
- Minutes on a $50 phone plan: $50 total budget. $35 is for the fixed monthly cost. $50 - $35 = $15 is the amount available for minutes. Since each minute costs $0.10, you can talk 15/0.10 = 150 minutes.
Square Root of a Negative Number
- Square root of -16: The square root of a negative number is an imaginary number. √(-16) = 4i
Multiplying Complex Numbers
- Multiplying (2+3i)(4-i): Use the distributive property (FOIL) to multiply. (2+3i)(4-i) = 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i
Solving Quadratic Equations
- Solving 2x^2 + 5x - 3 = 0 using the quadratic formula: Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. For this equation, a = 2, b = 5, and c = -3. Calculating this gives x = (-5 ± √(25 + 24)) / 4. Result: x = 1/2 or x = -3.
Multiplying Polynomials
- Multiplying (x+2)(x^2-3x+1): Use the distributive property to multiply the polynomials. (x+2)(x^2-3x+1) = x^3 - 3x^2 + x + 2x^2 - 6x + 2 = x^3 - x^2 - 5x + 2
Polynomial Division (Synthetic Division)
- Dividing x^3 - 2x^2 + 4x - 1 by x-1 using synthetic division: Using synthetic division with the divisor (x-1), we get: 1 | 1 -2 4 -1 | 1 -1 3 ------------------ 1 -1 3 2 Thus, the result is x^2 - x + 3 with a remainder of 2.
Solving Polynomial Equations by Factoring
- Solving x^3 - 4x = 0 by factoring: Factor out x to get x(x^2 - 4) = 0. Then factor the difference of squares (x^2 - 4) into (x-2)(x+2). Solutions are x = 0, x = 2, and x = -2.
Number of Solutions
- Solutions for x^5 - 3x^3 + 2x = 0: The equation has 5 solutions because the highest power is 5.
Zeros of a Polynomial
- Possible zeros for f(x) = x^4 - 3x^3 + 2x^2 - x + 1: Descartes' Rule of Signs applies. By examining the signs of the coefficients, there is one possible positive real root. For negative real roots, there is no sign variation in the coefficients for f(-x) = x^4 + 3x^3 + 2x^2 + x + 1, which means no negative real roots are possible. Therefore, there are zero negative real roots or four positive real roots.
Polynomial Function Properties
- Leading coefficient, degree, and end behavior of f(x) = -2x^3 + 5x^2 - x + 7: Leading coefficient is -2, degree is 3, and the end behavior is that as x → ∞, f(x) → -∞ ; and as x → -∞, f(x) → ∞.
Descartes' Rule of Signs
- Sign changes in f(x) = x^3 - 2x^2 + 3x - 1: There are three sign changes.
Turning Points of a Polynomial
- Maximum number of turning points for f(x) = x^4 - 3x^2 + 2: The maximum number of turning points is (degree - 1) = 3
Determining Even/Odd Functions
- f(x) = x^3 - x is odd: A function is odd if f(-x) = -f(x).
Solving Quadratic Equations by Taking the Square Root
- Solving x^2 - 16 = 0 by taking the square root: Add 16 to both sides; x^2 = 16; take the square root of both sides, x = ±√16; result is x = 4 and x = -4
Solving Complex Equations
- Values of x and y for (2x + yi) = (3 - 4i): Equate real and imaginary components to get 2x = 3 and y = -4. Results are x = 3/2 and y = -4.
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Description
This quiz covers key concepts in polynomial operations including expanding, factoring, cubing, and adding polynomials. It also introduces practical applications such as calculating costs for phone plans. Test your understanding of these mathematical principles.