Algebra II - Polynomial Functions

Let $\small f(x)=x^2-10$ , $\small g(x)=x^3$ , and $\small h(x)=2x$ . What is $\small f(h(g(x)))$ ?

$\small 4x^6-10$

$\small 64x^6-10$

$\small 8x^6-240x^4+960x^2-8000$

$\small 6x^2-60$

$\small \small 2x^6-10$

Explanation

When solving functions within functions, we begin with the innermost function and work our way outwards. Therefore:

$\small h(g(x))=2(x^3)=2x^3$

and

$\small f(h(g(x)))=(2x^3)^2-10=4x^6-10$

Let $\small f(x)=\sqrt[3]{x^2}$ , $\small g(x)=x^3$ , and $\small h(x)=\sqrt[4]{x}$ . What is $\small f(g(h(x)))$ ?

$\small \sqrt{x}$

$\small \sqrt[3]{x}$

$\small \sqrt[4]{x}$

$\small \sqrt[3]{x^2}$

$\small \sqrt[4]{x^3}$

Explanation

This problem relies on our knowledge of a radical expression $\small \sqrt[b]{x^a}$ equal to $\small x^\frac{a}{b}$ . The functions are subbed into one another in order from most inner to most outer function.

$\small g(h(x))=(\sqrt[4]{x})^3=x^\frac{3}{4}$

and

$\small f(g(h(x)))=\sqrt[3]{(x^\frac{3}{4})^2}=\sqrt[3]{x^\frac{6}{4}}=x^\frac{6}{12}=x^\frac{1}{2}=\sqrt{x}$

True or false:

The polynomial $7x^{5} - 4x^{2} + 2x + 13$ has as a factor.

True

False

Explanation

The easiest way to answer this question is arguably as follows:

Let $P(x) =7x^{5} - 4x^{2} + 2x + 13$ . By a corollary of the Factor Theorem, is divisible by if and only if the alternating sum of its coefficients (accounting for minus symbols) is 0.

To find this alternating sum, it is necessary to reverse the symbol before all terms of odd degree. In , there are two such terms - the fifth-degree and first-degree (linear) term, so alternating coefficient sum is

It follows that $7x^{5} - 4x^{2} + 2x + 13$ is divisible by .

If , find .

Explanation

Substitute for in the original equation:

Use FOIL or the Square of a Binomial Rule to find .

Recall that FOIL stands for the multiplication between the First components in both binomials followed by the Outer components, then the Inner components, and lastly the Last components.

Then, Distribute: .

Combine like terms to simplify:

If , find .

Explanation

Substitute 5y in for every x:

Simplify:

Square the first term:

Distribute the coefficients:

Find the product:

$6x^{2} + 14x +4$

$5x^{2}+12x+4$

$2x^{2}+3x+14$

$x^{2}+10x+3$

$x^{2}+12x+14$

Explanation

Using the FOIL (first, outer, inner, last) method, you can expand the polynomial to get

first: $2x\cdot 3x=6x^2$

outer: $2x\cdot 1=2x$

inner: $4x\cdot 3=12x$

lasts: $4\cdot 1=4$

From here, combine the like terms.

$6x^{2}+14x+4$

Factorize:

Explanation

In order to factorize this quadratic, we will need to identify the roots of the first and last term and order it into the two binomials.

We know that it will be in the form of:

The value of can be divided into , and is the only possibility to be replaced with and .

Substitute this into the binomials.

Now we need to determine $B\times D$ such that it will equal to 12, and satisfy the central term of .

The roots of 12 that can be interchangeable are:

The only terms that are possible are since

Remember that we must have a positive ending term!

This means that .

Substitute the terms.

The answer is:

The highest- and lowest-degree terms of a polynomial of degree 8 are $4x^{8}$ and , respectively; the polynomial has only integer coefficients.

True or false: By the Rational Zeroes Theorem, it is impossible for $\frac{9}{2}$ to be a zero of this polynomial.

True

False

Explanation

By the Rational Zeroes Theorem (RZT), if a polynomial has only integer coefficients, then any rational zero must be the positive or negative quotient of a factor of the constant and a factor of the coefficient of greatest degree. These integers are, respectively, 24, which as as its factors 1, 2, 3, 4, 6, 8, 12, and 24, and 4, which has as its factors 1, 2, and 4.

The complete set of quotients of factors of the former and factors of the latter is derived by dividing each element of $\left { 1, 2, 3, 4, 6, 8, 12, 24 \right }$ by each element of $\left { 1, 2, 4 \right }$ . The resulting set is

$\left { \pm \frac{1}{4}, \pm\frac{1}{2}, \pm \frac{3}{4}, \pm 1, \pm\frac{3}{2} , \pm2, \pm3, \pm4, \pm 6, \pm 8, \pm12, \pm 24 \right }$ ,

so any rational zero must be an element of this set. $\frac{9}{2}$ is not an element of this set, so by the RZT, it cannot be a zero of the polynomial.

True or false:

The polynomial $8x^{6}+ 7x^{4} - 15$ has as a factor.

True

False

Explanation

One way to answer this question is as follows:

Let $P(x) = 8x^{6}+ 7x^{4} - 15$ . By a corollary of the Factor Theorem, is divisible by if and only if the sum of its coefficients (accounting for minus symbols) is 0. $P(x) = 8x^{6}+ 7x^{4} - 15$ has

as its coefficient sum, so $8x^{6}+ 7x^{4} - 15$ is indeed divisible by .

A baseball is thrown off the roof of a building 220 feet high at an initial upward speed of 72 feet per second; the height of the baseball relative to the ground is modeled by the function

$h(t) = -16t^{2}+ 72t + 220$

How long does it take for the baseball to reach its highest point (nearest tenth of a second)?

$2.3\textrm{ sec}$

$6.6 \textrm{ sec}$

$2.1 \textrm{ sec}$

$8.4\textrm{ sec}$

$4.5\textrm{ sec}$

Explanation

The highest point of the ball is the vertex of the ball's parabolic path, so to find the number of seconds that is takes to reach this point, it is necessary to find the vertex of the parabola of the graph of the function