Without graphing the function f(x)=(2x+1)(x-4)^2 determine:
a) the degree of the function
b) the type of polynomial function
c) the leading coefficient
d) the zeroes
e) the y-intercept
f) the end behaviors
(a) x^1 * x^4 = x^5
(b) type? we have the degree, so what else is there?
I guess you could say "with real coefficients"
(c) 2*1^4 = ___
(d) if (x-a) is a factor, the x=a is a zero
(e) odd degree, with positive leading coefficient, so with x outside the interval containing all the roots, y<0 when x<0 and y>0 when x>0
expand it to see that the highest power is x^3, so the degree is 3
so it is a cubic function
the leading coefficient comes from the (2x+1) , so it is 2
the zeros are -1/2 and 4
for the y-intercept, let x = 0, then evaluate
for large values of x, (x-4)^2 is always +large, so is the 2x+1, so +large
for large negative values of x, (x-4)^2 is still +larg, but 2x+1 becomes negative lare, so - large
thanks, mathhelper
you were somewhat more careful than I was ...
Thank you both!
To answer these questions without graphing the function, we can look at the given expression and use the properties of polynomial functions.
a) The degree of a polynomial function is determined by the highest power of the variable in the expression. In this case, the highest power is 3 (because of the exponent 2 in (x-4)^2), so the degree of the function is 3.
b) The type of polynomial function is determined by the degree and the number of terms. The number of terms in this expression is 2 (because of (2x+1) and (x-4)^2). If the number of terms is 2, we have a binomial, and if it's more than 2, we have a polynomial with multiple terms. Since the degree is 3 and we have 2 terms, this function is a cubic binomial.
c) The leading coefficient is the coefficient of the term with the highest power. In this case, the term with the highest power is (x-4)^2, and its coefficient is 1. Therefore, the leading coefficient is 1.
d) To find the zeroes of the function, we set f(x) equal to zero and solve for x. So we have:
(2x+1)(x-4)^2 = 0
To find the zeroes, we set each factor equal to zero and solve:
2x + 1 = 0 -> x = -1/2
(x - 4)^2 = 0 -> x - 4 = 0 -> x = 4
Therefore, the zeroes of the function are x = -1/2 and x = 4.
e) The y-intercept is the value of the function when x is equal to zero. To find the y-intercept, we plug in x = 0 into the function:
f(0) = (2(0)+1)(0-4)^2 = (1)(16) = 16
Therefore, the y-intercept is 16.
f) The end behavior of a polynomial function is determined by the sign of the leading term (the term with the highest power) as x approaches positive and negative infinity.
In this case, the leading term is (x-4)^2. When x approaches positive infinity, (x-4)^2 approaches positive infinity because squaring a positive number yields a positive result. When x approaches negative infinity, (x-4)^2 approaches positive infinity as well since squaring a negative number also yields a positive result.
Therefore, the end behaviors of the function are both approaching positive infinity.