Complex Algebra! Please Help! I have been struggling really bad.
1. x4 – 3x2 + 2y – 1 is a polynomial of degree
A) 3
B) 2
C) 4
D) 5
2. For a given function ƒ(x), the translation ƒ(x – 2) + 3 means the function will shift
A) right 3 and upward 2.
B) left 3 and downward 2.
C) left 2 and downward –3.
D) right 2 and upward 3.
3. Which of the following describes the end behavior of the function ƒ(x) = x4 + 3x3 – 2x + 7?
A) As x → –∞, y → –∞ and as x → +∞, y → +∞
B) As x → –∞, y → –∞ and as x → +∞, y → –∞
C) As x → –∞, y → +∞ and as x → +∞, y → –∞
D) As x → –∞, y → +∞ and as x → +∞, y → +∞
4. Use Pascal's triangle to expand (x + 3)4.
A) x4 – 20x3 + 90x2 – 135x + 81
B) x4 – 12x3 + 54x2 – 108x + 81
C) x4 + 12x3 + 54x2 + 108x + 81
D) x4 + 20x3 + 90x2 + 135x + 81
5. Use Pascal's triangle to find the coefficient of the third term in the expansion of (a + b)2.
A) 2
B) 4
C) 1
D) 3
#1. The degree is the highest power of x -- x^4
#2. f(x-h)+k shifts right h, up k
#3. Since the highest term is +x^4, y is always positive for large x
#4. (x+3)^4 = x^4 + 4x^3*3 + 6x^2*3^2 + 4x*3^3 + 3^4
#5. (a+b)^2 = a^2 + 2ab + b^2
I'd be happy to help you with your complex algebra questions! Let's start with each question one by one:
1. To find the degree of a polynomial, we need to determine the highest exponent of any term in the polynomial. In this case, we have x^4 - 3x^2 + 2y - 1. The highest exponent is 4, so the polynomial has a degree of 4. Therefore, the answer is (C) 4.
2. The translation ƒ(x - 2) + 3 means that the function will shift horizontally by 2 units to the right (because of the "x - 2") and vertically by 3 units upward (because of the "+ 3"). So, the correct answer is (A) right 3 and upward 2.
3. The end behavior of a polynomial describes its behavior as x approaches positive or negative infinity. For the function ƒ(x) = x^4 + 3x^3 - 2x + 7, as x approaches negative infinity (x → -∞), the leading term x^4 dominates, and since x^4 is positive for all x, y approaches positive infinity (y → +∞). Similarly, as x approaches positive infinity (x → +∞), y also approaches positive infinity (y → +∞). Therefore, the correct answer is (A) As x → –∞, y → –∞ and as x → +∞, y → +∞.
4. Pascal's triangle can be used to expand binomial expressions. The fourth row of Pascal's triangle is 1, 4, 6, 4, 1, which corresponds to the coefficients of the terms in the expansion of (x + 3)^4.
Expanding (x + 3)^4 using Pascal's triangle gives us:
(x + 3)^4 = 1(x^4) + 4(x^3)(3) + 6(x^2)(3^2) + 4(x)(3^3) + 1(3^4)
= x^4 + 12x^3 + 54x^2 + 108x + 81
So, the correct answer is (C) x^4 + 12x^3 + 54x^2 + 108x + 81.
5. Pascal's triangle can be used to determine the coefficients in the expansion of (a + b)^2. The second row of Pascal's triangle is 1, 2, 1, which corresponds to the coefficients of the terms in the expansion of (a + b)^2.
Expanding (a + b)^2 using Pascal's triangle gives us:
(a + b)^2 = 1(a^2) + 2(a)(b) + 1(b^2)
= a^2 + 2ab + b^2
The coefficient of the third term (2ab) is 2. Therefore, the correct answer is (A) 2.
I hope this explanation helps you with your complex algebra questions! Let me know if you have any more questions.