1. Find the GCF of the polynomial. 8x^6+32x^3
2. Polynomial identities tell us that the square of the sum of a and b, (a+b)^2, can be rewritten as ______.
3. The sum of cubes identity is very powerful, although a little hard to remember. It tells us that a^3+b^3=______.
4. Factor the expression completely. 2x^2+4x-16
5. Is (x-4) a factor of (x^3+x^2-16x-16)?
6. What is the remainder when 2x^3+3x^2-x+1 is divided by x+3?
7. Describe the end behavior of the function. f(x)=2x^4-4x^2
8. What are the co-ordinates of the focus of the parabola y^2=12x?
9. What is the equation of the directrix of the parabola x^2=20y?
10. List the transformations to the parent function in y=-(x+4)^2-3
11. Find and select each real zero and its multiplicity. f(x)=x^2(x-2)^3(x-7)
1. To find the greatest common factor (GCF) of the polynomial 8x^6 + 32x^3, we need to determine the highest power of x that can be divided evenly by both terms. In this case, we can factor out 8x^3 from both terms, which gives us:
8x^6 + 32x^3 = 8x^3(x^3 + 4)
Therefore, the GCF of the polynomial is 8x^3.
2. The square of the sum of a and b, (a + b)^2, can be rewritten using the identity (a + b)^2 = a^2 + 2ab + b^2. This identity tells us that the square of the sum of two terms is equal to the sum of their squares plus twice their product.
3. The sum of cubes identity tells us that a^3 + b^3 can be factored as (a + b)(a^2 - ab + b^2). So, a^3 + b^3 = (a + b)(a^2 - ab + b^2).
4. To factor the expression 2x^2 + 4x - 16, we can look for two numbers that multiply to give the product of the coefficient of x^2 (2) and the constant term (-16), and add up to give the coefficient of x (4). In this case, the numbers are 8 and -2:
2x^2 + 4x - 16 = 2x^2 + 8x - 2x - 16
Now, we can factor by grouping:
(2x^2 + 8x) - (2x + 16) = 2x(x + 4) - 2(x + 4)
Factor out the common factor of (x + 4):
2(x + 4)(x - 2)
Therefore, the expression 2x^2 + 4x - 16 can be completely factored as 2(x + 4)(x - 2).
5. To determine whether (x - 4) is a factor of (x^3 + x^2 - 16x - 16), we can use synthetic division. Divide the polynomial by the potential factor:
4 | 1 1 -16 -16
--------------
1 5 4 0
Since the remainder is 0, (x - 4) is indeed a factor of (x^3 + x^2 - 16x - 16).
6. To find the remainder when 2x^3 + 3x^2 - x + 1 is divided by x + 3, we can use synthetic division. Divide the polynomial by the divisor:
-3 | 2 3 -1 1
-----------------
2 -3 8 -23
The remainder is -23. Therefore, the remainder when 2x^3 + 3x^2 - x + 1 is divided by x + 3 is -23.
7. To describe the end behavior of the function f(x) = 2x^4 - 4x^2, we examine the highest degree term and the coefficient of that term. In this case, the highest degree term is 2x^4 and the coefficient is 2.
As x approaches positive infinity, the term 2x^4 dominates the function. Since the coefficient is positive, the function will approach positive infinity as x increases without bound. Similarly, as x approaches negative infinity, the function will also approach positive infinity.
Therefore, the end behavior of the function f(x) = 2x^4 - 4x^2 is that as x approaches positive or negative infinity, f(x) approaches positive infinity.
8. The coordinates of the focus of the parabola y^2 = 12x can be found using the formula (h, k + 1/(4a)), where (h, k) is the vertex of the parabola and "a" is the coefficient of x. In this case, "a" is 12, so the focus coordinates can be calculated as follows:
h = 0 (since the parabola is symmetrical with respect to the y-axis)
k = 0 (since the vertex is at the origin)
a = 12
Using the formula, we get the focus coordinates as (0, 1/(4×12)) or (0, 1/48).
Therefore, the coordinates of the focus of the parabola y^2 = 12x are (0, 1/48).
9. The equation of the directrix of the parabola x^2 = 20y can be found using the formula y = -k - 1/(4a), where (h, k) is the vertex of the parabola and "a" is the coefficient of y. In this case, "a" is 5, so the directrix equation can be calculated as follows:
h = 0 (since the parabola is symmetrical with respect to the y-axis)
k = 0 (since the vertex is at the origin)
a = 5
Using the formula, we get the directrix equation as y = -0 - 1/(4×5) or y = -1/20.
Therefore, the equation of the directrix of the parabola x^2 = 20y is y = -1/20.
10. In the function y = -(x + 4)^2 - 3, there are several transformations applied to the parent function y = x^2.
A. Reflection: The negative sign in front of the function reflects the graph vertically, resulting in an upside-down parabola.
B. Horizontal Translation: The term (x + 4) inside the parentheses shifts the graph 4 units to the left. The opposite sign indicates a rightward shift for positive values.
C. Vertical Translation: The -3 at the end of the function shifts the graph downward by 3 units.
Therefore, the transformations to the parent function y = x^2 in y = -(x + 4)^2 - 3 are a reflection, a horizontal translation of 4 units to the left, and a vertical translation of 3 units downward.
11. To find the real zeros and their multiplicities for the function f(x) = x^2(x - 2)^3(x - 7), we set each factor equal to zero and solve for x:
x^2 = 0
x = 0 (multiplicity 2)
(x - 2) = 0
x = 2 (multiplicity 1)
(x - 7) = 0
x = 7 (multiplicity 1)
Therefore, the real zeros of f(x) = x^2(x - 2)^3(x - 7) are x = 0 (with multiplicity 2), x = 2, and x = 7.
Ahhh! A Homework dump. Well, here are some hints.
#1. Note that 8x^3 divides each term
#2,3: you must have these identities in your text. If not, google sum of cubes
#4. 2(x^2+2x-8) ... now, which two factors of 8 differ by 2?
#5,6 Use the Remainder Theorem
#7 It's a polynomial of even degree, so f(x) → +∞ for both ±x
#8,9 y^2 = 2px has focus at (p,0) and directrix at x = -p
#10 f(x-h) shifts right by h f(x)+k shifts up by k
#11 the zeroes are the values of x that make each factor 0
The multiplicities are the powers of each root.