Question

algebra 2-17411200330a106
cumulative assessment 2
due dec 19 - 3:00 pm

consider the polynomial function ( f(x) = x^3 + 2x^2 + x^4 - 8 + 4x ).
select all the statements that are true about the function.

a. the degree of the polynomial is 3.

b. written in standard form the function is ( f(x) = x^4 + x^3 + 2x^2 + 4x - 8 ).

c. as ( x ) increases or decreases, ( f(x) ) increases.

d. the polynomial function has a maximum.

e. the ( x )-intercepts are at ( x = -1 ) and ( x = 2 ).

f. the function is negative over the interval ( (-2, 1) ).

Explanation:

Step1: Analyze Option A

The degree of a polynomial is the highest power of \(x\). For \(f(x)=x^{4}+x^{3}+2x^{2}+4x - 8\) (after re - ordering), the highest power is \(4\), not \(3\). So A is false.

Step2: Analyze Option B

To write a polynomial in standard form, we arrange the terms in descending order of the exponents of \(x\). Given \(f(x)=x^{3}+2x^{2}+x^{4}-8 + 4x\), re - ordering gives \(f(x)=x^{4}+x^{3}+2x^{2}+4x - 8\). So B is true.

Step3: Analyze Option C

For a polynomial \(y = ax^{n}+\cdots\) with \(n\) even and \(a>0\) (here \(n = 4\) and \(a = 1>0\)), as \(x\to\pm\infty\), \(y\to+\infty\). But when \(x\) is a large negative number, we can't say that \(f(x)\) is always increasing as \(x\) decreases. The end - behavior is that as \(x\to-\infty\), \(f(x)\to+\infty\) and as \(x\to+\infty\), \(f(x)\to+\infty\), but the function has a minimum (since the leading coefficient is positive and degree is even), not that it is always increasing as \(x\) increases or decreases. So C is false.

Step4: Analyze Option D

A polynomial of even degree with a positive leading coefficient (here degree \(4\), leading coefficient \(1>0\)) has a minimum, not a maximum. So D is false.

Step5: Analyze Option E

To find the \(x\) - intercepts, we set \(f(x)=0\), i.e., \(x^{4}+x^{3}+2x^{2}+4x - 8=0\). Let's test \(x=-1\): \((-1)^{4}+(-1)^{3}+2(-1)^{2}+4(-1)-8=1 - 1 + 2-4 - 8=-10
eq0\). Test \(x = 2\): \(2^{4}+2^{3}+2\times2^{2}+4\times2-8=16 + 8+8 + 8-8=32
eq0\). So E is false.

Step6: Analyze Option F

Let's test a value in the interval \((-2,1)\), say \(x = 0\). Then \(f(0)=0^{4}+0^{3}+2\times0^{2}+4\times0-8=-8<0\). Test \(x=-1\): \(f(-1)=(-1)^{4}+(-1)^{3}+2(-1)^{2}+4(-1)-8=1 - 1+2 - 4 - 8=-10<0\). Test \(x = 0.5\): \(f(0.5)=(0.5)^{4}+(0.5)^{3}+2\times(0.5)^{2}+4\times(0.5)-8=0.0625 + 0.125+0.5 + 2-8=-5.3125<0\). So the function is negative over the interval \((-2,1)\), F is true.

Answer:

B. Written in standard form the function is \(f(x)=x^{4}+x^{3}+2x^{2}+4x - 8\), F. The function is negative over the interval \((-2,1)\)