1) if the polynomial p(x) has factories of 12, (x-5), and (x+4), which of the following must also be a factor of p(x)?
2x^2 + 8
4x^2-20
6x^2 -6x-120
x^2-10x+25
2) IF f(x)= -x+7 and g(f(x))= 2x+1, what is the value of g(2)?
^I got 11
The square of a positive number is .24 greater than the number itself. What is that number?
Hmmm. Never heard of polynomial factories. So, assuming you meant factors,
If (x-5) and (x+4) are factors, so is
(x-5)(x+4) = x^2-x-20
That's not one of the choices, but if 12 is also a factor, so is 6. Now what do you say?
since f(5) = 2,
g(2) = g(f(5)) = 2*5+1 = 11
good job
x^2 = x + .24
x^2-x-.24 = 0
x = 1.2
check: 1.2^2 = 1.44 = 1.2+.24
polynomials have factors, not factories ....
we could write the factors of p(x) as
2*2*3*(x-5)*(x+4)
any combination of these will also be a factor.
How about 2*3*(x-5)(x+4)
expand it and see what you get
2) correct
3)
x^2 - x = .24
x^2 - x -.24 = 0
solve for x using the quadratic formula
1) To find out which of the given polynomials must also be a factor of p(x), we can use the factor theorem. According to the factor theorem, if a polynomial p(x) has a factor (x - a), then p(a) must be equal to zero.
Let's substitute the given factors into p(x) and check if they are equal to zero:
For the factor (x - 5), substituting 5 into p(x) should give us zero:
p(5) = 0 if and only if (5 - 5) = 0, which is true.
For the factor (x + 4), substituting -4 into p(x) should give us zero:
p(-4) = 0 if and only if (-4 + 4) = 0, which is true.
However, we don't have the exact polynomial p(x), so we need to check each answer option by substituting the corresponding values into the polynomial.
2x^2 + 8: Substituting x = 5, we get (2*5^2 + 8) = 58, which is not equal to zero.
4x^2 - 20: Substituting x = -4, we get (4*(-4)^2 - 20) = 80, which is not equal to zero.
6x^2 - 6x - 120: Substituting x = 5, we get (6*5^2 - 6*5 - 120) = 30, which is not equal to zero.
x^2 - 10x + 25: Substituting x = -4, we get ((-4)^2 - 10*(-4) + 25) = 57, which is not equal to zero.
Based on this analysis, none of the given options must be a factor of p(x).
2) To find the value of g(2), we need to substitute x = 2 into the expression g(f(x)) = 2x + 1.
First, let's find the value of f(x) when x = 2:
f(2) = -(2) + 7 = 5
Now we can substitute f(2) = 5 into g(f(x)):
g(f(2)) = g(5) = 2(5) + 1 = 11
Therefore, the value of g(2) is indeed 11.
3) Let's represent the number we are looking for as x. According to the given information, the square of x is 0.24 greater than x itself. We can write this as an equation:
x^2 = x + 0.24
Now, let's rearrange the equation to solve for x:
x^2 - x - 0.24 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula. In this case, factoring might not be straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation (x^2 - x - 0.24 = 0), the values of a, b, and c are:
a = 1, b = -1, c = -0.24
Substituting these values into the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(1)(-0.24))) / (2(1))
x = (1 ± √(1 + 0.96)) / 2
x = (1 ± √(1.96)) / 2
Using a calculator, we find that the two solutions are approximately:
x = 1.240, x = -0.240
Since we're looking for a positive number, the solution is x = 1.240.