Question 1

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$x ^ { 5 } y ^ { 2 } - 6 x ^ { 4 } y ^ { 3 } - 16 x ^ { 3 } y ^ { 4 }$

Accepted Answer

The common factor in all terms is $x^3y^2$, so we can factor that out to get $x^3y^2(x-8y-2y)$. Simplifying the expression in the parentheses gives us $x-10y$ which is the only choice that matches this factorization. Therefore, the answer is choice C.

Question 2

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x.
-The difference of two positive numbers is 6 and their product is 187. Find the two numbers

Accepted Answer

The answer of Find all fixed points for the function...

Question 3

Solve the problem.
-Let $f ( x ) = 3 x ^ { 2 } - 18 x + 24$. Find all values $x$ for which $f ( x ) = 0$.

Accepted Answer

To find the values of $x$ for which $f(x) = 0$, we set the given function equal to zero and solve for $x$: $3x^2 - 18x + 24 = 0$. Factoring out the common factor of 3 gives $x^2 - 6x + 8 = 0$. Factoring the quadratic equation further, we get $(x-2)(x-4) = 0$, which gives the solutions $x = 2$ and $x = 4$.

Question 4

Solve the problem.
-Let $f ( x ) = - x ^ { 2 } + 4 x$. Find all values $x$ for which $f ( x ) = - 21$.

Accepted Answer

We need to solve for $x$ when $f(x)=-21$, so we have $$-x^2+4x=-21.$$ Simplifying,
$$x^2-4x+21=0.$$ Applying the quadratic formula, we have $$x=\frac{4\pm\sqrt{16-4(1)(21)}}{2(1)}=\frac{4\pm\sqrt{-68}}{2}=2\pm i\sqrt{17}.$$ Since $x$ is a real value, the only possible values are $2+\sqrt{17}$ or $2-\sqrt{17}$ (Note that $-3+0i$ and $7+0i$ are conjugates, so they will give the same value), which are $\boxed{B) 7,-3}$.

Question 5

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$3 - x ^ { 2 } - 72$

Accepted Answer

The given polynomial is $3 - x^2 - 72$, which simplifies to $-x^2 - 69$. This expression cannot be factored into a product of polynomials with real coefficients, as it does not fit the form of a difference of squares or any other factorable pattern.

Question 6

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x.
-One positive number is 6 less than twice a second number, and their product is 140. Find the two numbers.

Accepted Answer

The answer of Find all fixed points for the function...

Question 7

Solve the problem.
-Let $f ( x ) = x ^ { 2 } + 12 x + 29$. Find all values $x$ for which $f ( x ) = 2$.

Accepted Answer

We need to solve for $x$ such that $f(x)=2$:
\begin{align*}
x^2+12x+29&=2 \
x^2+12x+27&=0 \
(x+9)(x+3)&=0
\end{align*}
Therefore, the solutions are $x=-9$ and $x=-3$, which corresponds to choice C).

Question 8

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$$f ( x ) = 4 x ^ { 2 } + 3 x - 5 , f ( a - 1 )$$

Accepted Answer

The answer of Factor completely. If the polynomial cannot be...

Question 9

Solve the problem.
-Let $f ( x ) = x ^ { 2 } + x - 12$. Find all values $x$ for which $f ( x ) = 0$.

Accepted Answer

The quadratic equation $x^2 + x - 12 = 0$ can be factored as $(x - 3)(x + 4) = 0$. Solving for $x$, we get $x = 3$ and $x = -4$.

Question 10

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x.
-$$f ( x ) = x ^ { 2 } - 5 x - 55$$

Accepted Answer

To find fixed points, solve $x^2 - 5x - 55 = x$. Rearrange to $x^2 - 6x - 55 = 0$. Solving this quadratic equation gives roots $x = -5$ and $x = 11$.

Question 11

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$$16 x ^ { 2 } - 24 x + 9$$

Accepted Answer

This polynomial is a perfect square trinomial. We can factor it using the formula $$(a \pm b) ^ 2= a^2 \pm 2ab + b^2$$ with $a = 4x$ and $b = \frac{3}{2}$. Then, $$(4x - 3) ^ 2 = (4x)^2 - 2 \cdot 4x \cdot 3 + (3) ^ 2 = 16 x ^ 2 - 24x + 9$$ Since there are no other factors, the answer is (A).

Question 12

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x.
-$$f ( x ) = x ^ { 2 } - 6 x + 10$$

Accepted Answer

To find the fixed points, set $f(x) = x$, which gives us the equation $x^2 - 6x + 10 = x$. Simplifying, we get $x^2 - 7x + 10 = 0$. Factoring, we find $(x - 5)(x - 2) = 0$, which gives us the fixed points $x = 5$ and $x = 2$.

Question 13

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$$3 x ^ { 2 } + 3 x - 18$$

Accepted Answer

We can factor out a 3 first to get
$$3(x^2 + x - 6)$$
Then, we can factor the quadratic expression inside the parentheses
$$3(x+3)(x-2)$$
Therefore, the polynomial completely factors to $$3(x+3)(x-2)$$ which is choice B.

Question 14

Solve the problem.
-Let $f ( x ) = x ^ { 2 } + 15 x$. Find all values $x$ for which $f ( x ) = - 54$.

Accepted Answer

To find the values of $x$ for which $f(x) = -54$, we substitute $-54$ for $f(x)$ in the given equation and solve for $x$:$$x^2 + 15x = -54$$$$x^2 + 15x + 54 = 0$$Factoring the quadratic equation, we get:$$(x + 9)(x + 6) = 0$$Setting each factor equal to zero gives us the solutions:$$x + 9 = 0 \Rightarrow x = -9$$$$x + 6 = 0 \Rightarrow x = -6$$Therefore, the correct values of $x$ are $-9$ and $-6$.

Question 15

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$$x ^ { 2 } + 6 x y + 9 y ^ { 2 }$$

Accepted Answer

This is a perfect square trinomial, which means it can be factored as $(x+3y)^2$.

Question 16

Solve the problem.
-Let $f ( x ) = x ^ { 2 } - x - 56$. Find all values $x$ for which $f ( x ) = 0$.

Accepted Answer

To find the values of $x$ for which $f(x) = 0$, we set the equation $x^2 - x - 56 = 0$ and solve for $x$. Factoring the quadratic equation, we get $(x - 8)(x + 7) = 0$, which gives us the solutions $x = 8$ and $x = -7$.

Question 17

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x.
-The product of the page numbers on two facing pages of a book is 506. Find the page numbers.

Accepted Answer

The answer of Find all fixed points for the function...

Question 18

Solve the problem.
-Let $f ( x ) = x ^ { 2 } + 3 x - 108$. Find all values $x$ for which $f ( x ) = 0$.

Accepted Answer

To find the values of $x$ for which $f(x) = 0$, we solve the equation $x^2 + 3x - 108 = 0$. Factoring the quadratic equation, we get $(x + 12)(x - 9) = 0$, which gives us the solutions $x = -12$ and $x = 9$.

Question 19

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$$24 x ^ { 6 } + 66 x ^ { 4 } + 45 x ^ { 2 }$$

Accepted Answer

First, factor out the greatest common factor, which is $3x^2$, giving $3x^2(8x^4 + 22x^2 + 15)$. Then, factor the trinomial inside the parenthesis to get $3x^2(4x^2 + 5)(2x^2 + 3)$.

Question 20

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ
-$$- x ^ { 2 } - 2 x + 80$$

Accepted Answer

The answer of Factor completely. If the polynomial cannot be...

Quiz 6: Factoring and Quadratic Equations

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $x ^ { 5 } y ^ { 2 } - 6 x ^ { 4 } y ^ { 3 } - 16 x ^ { 3 } y ^ { 4 }$

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. -The difference of two positive numbers is 6 and their product is 187. Find the two numbers

Solve the problem. -Let $f ( x ) = 3 x ^ { 2 } - 18 x + 24$ . Find all values $x$ for which $f ( x ) = 0$ .

Solve the problem. -Let $f ( x ) = - x ^ { 2 } + 4 x$ . Find all values $x$ for which $f ( x ) = - 21$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $3 - x ^ { 2 } - 72$

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. -One positive number is 6 less than twice a second number, and their product is 140. Find the two numbers.

Solve the problem. -Let $f ( x ) = x ^ { 2 } + 12 x + 29$ . Find all values $x$ for which $f ( x ) = 2$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $f ( x ) = 4 x ^ { 2 } + 3 x - 5 , f ( a - 1 )$

Solve the problem. -Let $f ( x ) = x ^ { 2 } + x - 12$ . Find all values $x$ for which $f ( x ) = 0$ .

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. - $f ( x ) = x ^ { 2 } - 5 x - 55$

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $16 x ^ { 2 } - 24 x + 9$

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. - $f ( x ) = x ^ { 2 } - 6 x + 10$

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $3 x ^ { 2 } + 3 x - 18$

Solve the problem. -Let $f ( x ) = x ^ { 2 } + 15 x$ . Find all values $x$ for which $f ( x ) = - 54$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $x ^ { 2 } + 6 x y + 9 y ^ { 2 }$

Solve the problem. -Let $f ( x ) = x ^ { 2 } - x - 56$ . Find all values $x$ for which $f ( x ) = 0$ .

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. -The product of the page numbers on two facing pages of a book is 506. Find the page numbers.

Solve the problem. -Let $f ( x ) = x ^ { 2 } + 3 x - 108$ . Find all values $x$ for which $f ( x ) = 0$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $24 x ^ { 6 } + 66 x ^ { 4 } + 45 x ^ { 2 }$

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $- x ^ { 2 } - 2 x + 80$

Quiz 6: Factoring and Quadratic Equations

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - x5y2−6x4y3−16x3y4x ^ { 5 } y ^ { 2 } - 6 x ^ { 4 } y ^ { 3 } - 16 x ^ { 3 } y ^ { 4 }x5y2−6x4y3−16x3y4

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. -The difference of two positive numbers is 6 and their product is 187. Find the two numbers

Solve the problem. -Let f(x) =3x2−18x+24f ( x ) = 3 x ^ { 2 } - 18 x + 24f(x) =3x2−18x+24 . Find all values xxx for which f(x) =0f ( x ) = 0f(x) =0 .

Solve the problem. -Let f(x) =−x2+4xf ( x ) = - x ^ { 2 } + 4 xf(x) =−x2+4x . Find all values xxx for which f(x) =−21f ( x ) = - 21f(x) =−21 .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - 3−x2−723 - x ^ { 2 } - 723−x2−72

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. -One positive number is 6 less than twice a second number, and their product is 140. Find the two numbers.

Solve the problem. -Let f(x) =x2+12x+29f ( x ) = x ^ { 2 } + 12 x + 29f(x) =x2+12x+29 . Find all values xxx for which f(x) =2f ( x ) = 2f(x) =2 .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - f(x) =4x2+3x−5,f(a−1) f ( x ) = 4 x ^ { 2 } + 3 x - 5 , f ( a - 1 ) f(x) =4x2+3x−5,f(a−1)

Solve the problem. -Let f(x) =x2+x−12f ( x ) = x ^ { 2 } + x - 12f(x) =x2+x−12 . Find all values xxx for which f(x) =0f ( x ) = 0f(x) =0 .

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. - f(x) =x2−5x−55f ( x ) = x ^ { 2 } - 5 x - 55f(x) =x2−5x−55

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - 16x2−24x+916 x ^ { 2 } - 24 x + 916x2−24x+9

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. - f(x) =x2−6x+10f ( x ) = x ^ { 2 } - 6 x + 10f(x) =x2−6x+10

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - 3x2+3x−183 x ^ { 2 } + 3 x - 183x2+3x−18

Solve the problem. -Let f(x) =x2+15xf ( x ) = x ^ { 2 } + 15 xf(x) =x2+15x . Find all values xxx for which f(x) =−54f ( x ) = - 54f(x) =−54 .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - x2+6xy+9y2x ^ { 2 } + 6 x y + 9 y ^ { 2 }x2+6xy+9y2

Solve the problem. -Let f(x) =x2−x−56f ( x ) = x ^ { 2 } - x - 56f(x) =x2−x−56 . Find all values xxx for which f(x) =0f ( x ) = 0f(x) =0 .

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. -The product of the page numbers on two facing pages of a book is 506. Find the page numbers.

Solve the problem. -Let f(x) =x2+3x−108f ( x ) = x ^ { 2 } + 3 x - 108f(x) =x2+3x−108 . Find all values xxx for which f(x) =0f ( x ) = 0f(x) =0 .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - 24x6+66x4+45x224 x ^ { 6 } + 66 x ^ { 4 } + 45 x ^ { 2 }24x6+66x4+45x2

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - −x2−2x+80- x ^ { 2 } - 2 x + 80−x2−2x+80

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $x ^ { 5 } y ^ { 2 } - 6 x ^ { 4 } y ^ { 3 } - 16 x ^ { 3 } y ^ { 4 }$

Solve the problem. -Let $f ( x ) = 3 x ^ { 2 } - 18 x + 24$ . Find all values $x$ for which $f ( x ) = 0$ .

Solve the problem. -Let $f ( x ) = - x ^ { 2 } + 4 x$ . Find all values $x$ for which $f ( x ) = - 21$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $3 - x ^ { 2 } - 72$

Solve the problem. -Let $f ( x ) = x ^ { 2 } + 12 x + 29$ . Find all values $x$ for which $f ( x ) = 2$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $f ( x ) = 4 x ^ { 2 } + 3 x - 5 , f ( a - 1 )$

Solve the problem. -Let $f ( x ) = x ^ { 2 } + x - 12$ . Find all values $x$ for which $f ( x ) = 0$ .

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. - $f ( x ) = x ^ { 2 } - 5 x - 55$

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $16 x ^ { 2 } - 24 x + 9$

Find all fixed points for the function by setting the function equal to x and solving the resulting equation for x. - $f ( x ) = x ^ { 2 } - 6 x + 10$

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $3 x ^ { 2 } + 3 x - 18$

Solve the problem. -Let $f ( x ) = x ^ { 2 } + 15 x$ . Find all values $x$ for which $f ( x ) = - 54$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $x ^ { 2 } + 6 x y + 9 y ^ { 2 }$

Solve the problem. -Let $f ( x ) = x ^ { 2 } - x - 56$ . Find all values $x$ for which $f ( x ) = 0$ .

Solve the problem. -Let $f ( x ) = x ^ { 2 } + 3 x - 108$ . Find all values $x$ for which $f ( x ) = 0$ .

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $24 x ^ { 6 } + 66 x ^ { 4 } + 45 x ^ { 2 }$

Factor completely. If the polynomial cannot be factored, write ʺprime.ʺ - $- x ^ { 2 } - 2 x + 80$