Question 1

Perform the division.
-$\left(x^{4}-625\right) \div(x+5)$
A) $x^{3}-5 x^{2}+25 x-125$
B) $x^{2}+5 x+25$
C) $x^{3}+5 x^{2}-25 x+125$
D) $x^{3}+5 x^{2}+25 x+125$

Accepted Answer

The given expression $\left(x^{4}-625\right) \div(x+5)$ can be simplified by recognizing that $x^4 - 625$ is a difference of squares, where $x^4 = (x^2)^2$ and $625 = 25^2$. Thus, $x^4 - 625 = (x^2 - 25)(x^2 + 25)$. Further simplifying, $x^2 - 25$ can also be factored as a difference of squares, yielding $(x - 5)(x + 5)$. However, since we are dividing by $x + 5$, we only need to consider the quotient when $x^4 - 625$ is divided by $x + 5$, which simplifies directly to $x^3 - 5x^2 + 25x - 125$, without needing to explicitly factor the expression as described. This is a standard result from polynomial long division or synthetic division.

Question 2

Perform the division.
-$\frac{x^{4}+3 x^{2}+10}{x^{2}+1}$
A) $x^{2}+2+\frac{7}{2}$
B) $x^{2}+2$
C) $x^{2}+2 x+4$
D) $x^{2}+2+\frac{8}{x^{2}+1}$

Accepted Answer

Dividing $x^{4}+3x^{2}+10$ by $x^{2}+1$ gives $x^{2}+2$ with a remainder of $8$, so the correct expression is $x^{2}+2+\frac{8}{x^{2}+1}$.

Question 3

Perform the division.
-$\frac{\mathrm{x}^{8}-1}{\mathrm{x}^{4}-1}$
A) $x^{4}+12 x-1$
B) $x^{4}-1$
C) $x^{4}$
D) $x^{4}+1$

Accepted Answer

The division $\frac{x^8 - 1}{x^4 - 1}$ can be simplified by recognizing both the numerator and the denominator as difference of squares. $x^8 - 1 = (x^4)^2 - 1^2$ and $x^4 - 1 = (x^2)^2 - 1^2$, which allows us to use the formula $a^2 - b^2 = (a - b)(a + b)$. Applying this to both the numerator and the denominator, we get $\frac{(x^4 - 1)(x^4 + 1)}{x^4 - 1}$. The $x^4 - 1$ terms cancel out, leaving $x^4 + 1$.

Question 4

Perform the division.
-$\frac{-16 x^{3}-4 x^{2}-6 x-9}{4 x+3}$
A) $x^{2}-2 x+3$
B) $-4 x^{2}-3$
C) $-4 x^{2}+2 x-3$
D) $x^{2}+2 x-3$

Accepted Answer

To divide $\frac{-16 x^{3}-4 x^{2}-6 x-9}{4 x+3}$, we use polynomial long division or synthetic division. The first term of the quotient is obtained by dividing $-16x^3$ by $4x$, which gives $-4x^2$. Multiplying $4x+3$ by $-4x^2$ and subtracting from the original polynomial, we proceed to the next terms similarly, ending up with $-4 x^{2}+2 x-3$ as the quotient.

Question 5

Perform the division.
-$\frac{12 x^{3}+3 x^{2}-35 x-13}{-4 x-5}$
A) $-3 x^{2}+3 x+5+\frac{15}{-4 x-5}$
B) $-3 x^{2}+3 x+5+\frac{12}{-4 x-5}$
C) $-3 x^{2}+3 x+5$
D) $x^{2}+5+\frac{3}{-4 x-5}$

Accepted Answer

The division of the polynomial $12x^3 + 3x^2 - 35x - 13$ by $-4x - 5$ results in the quotient $-3x^2 + 3x + 5$ with a remainder of 12, which is expressed as $\frac{12}{-4x-5}$.

Question 6

Provide an appropriate response.
-$\left(\frac{\mathrm{r}}{\mathrm{s}}\right)^{4}=\frac{\mathrm{r}^{4}}{\mathrm{~s}}$

Accepted Answer

The correct simplification is $\left(\frac{\mathrm{r}}{\mathrm{s}}\right)^{4}=\frac{\mathrm{r}^{4}}{\mathrm{s}^{4}}$.

Question 7

Write the word or phrase that best completes each statement or answers the question.
-Simplify the expressions $(2 x)^{0}$ and $\left(2 x^{0}\right)$ and explain how you arrived at your answers.

Accepted Answer

The answer of Write the word or phrase that best...

Question 8

Write the word or phrase that best completes each statement or answers the question.
-When is the square of the sum the sum of a square; i.e., when is $(A+B)^{2}=A^{2}+B^{2}$ ?

Accepted Answer

The answer of Write the word or phrase that best...

Question 9

Write the word or phrase that best completes each statement or answers the question.
-What must be done to the dividend before beginning the "long division" process for $\left(6 x^{2}+9 x^{4}+3 x^{3}+4 x+8\right) \div(6 x+8) ?$

Accepted Answer

The answer of Write the word or phrase that best...

Question 10

Choose the one alternative that best completes the statement or answers the question.
-Rank the numbers in order from smallest to langest.
$5)62 \times 10^{3}, 7.22 \times 10^{5}, 4.62 \times 10^{-2}, 7.90 \times 10^{3}, 5.04 \times 10^{-5}$
A) $4.62 \times 10^{-2}<5.04 \times 10^{-5}<5.62 \times 10^{3}<7.90 \times 10^{3}<$ $7.22 \times 10^{5}$
B) $7.22 \times 10^{5}<7.90 \times 10^{3}<5.62 \times 10^{3}<4.62 \times 10^{-2}<$ $5.04 \times 10^{-5}$
C) $5.04 \times 10^{-5}<5.62 \times 10^{3}<7.90 \times 10^{3}<7.22 \times 10^{5}<$ $4.62 \times 10^{-2}$
D) $5.04 \times 10^{-5}<4.62 \times 10^{-2}<5.62 \times 10^{3}<7.90 \times 10^{3}<$ $7.22 \times 10^{5}$

Accepted Answer

The correct order from smallest to largest is based on the exponent values and the base numbers for those with the same exponent. Negative exponents indicate numbers less than one, and the larger the positive exponent, the larger the number.

Question 11

Choose the one alternative that best completes the statement or answers the question.
-Tell whether or not the number is given in scientific notation. $5.240 \times 10^{-7}$

Accepted Answer

The number $5.240 \times 10^{-7}$ is in scientific notation because it is written as a product of a number between 1 and 10 and a power of 10.

Question 12

Choose the one alternative that best completes the statement or answers the question.
-Tell whether or not the number is given in scientific notation. $78.92 \times 10^{6}$

Accepted Answer

In scientific notation, the number should be between 1 and 10, so it should be written as $7.892 \times 10^{7}$.

Quiz 4: Polynomials: Operations

Perform the division. - $\left(x^{4}-625\right) \div(x+5)$

Perform the division. - $\frac{x^{4}+3 x^{2}+10}{x^{2}+1}$

Perform the division. - $\frac{\mathrm{x}^{8}-1}{\mathrm{x}^{4}-1}$

Perform the division. - $\frac{-16 x^{3}-4 x^{2}-6 x-9}{4 x+3}$

Perform the division. - $\frac{12 x^{3}+3 x^{2}-35 x-13}{-4 x-5}$

Provide an appropriate response. - $\left(\frac{\mathrm{r}}{\mathrm{s}}\right)^{4}=\frac{\mathrm{r}^{4}}{\mathrm{~s}}$

Write the word or phrase that best completes each statement or answers the question. -Simplify the expressions $(2 x)^{0}$ and $\left(2 x^{0}\right)$ and explain how you arrived at your answers.

Write the word or phrase that best completes each statement or answers the question. -When is the square of the sum the sum of a square; i.e., when is $(A+B)^{2}=A^{2}+B^{2}$ ?

Write the word or phrase that best completes each statement or answers the question. -What must be done to the dividend before beginning the "long division" process for $\left(6 x^{2}+9 x^{4}+3 x^{3}+4 x+8\right) \div(6 x+8) ?$

Choose the one alternative that best completes the statement or answers the question. -Rank the numbers in order from smallest to langest. $5) 62 \times 10^{3}, 7.22 \times 10^{5}, 4.62 \times 10^{-2}, 7.90 \times 10^{3}, 5.04 \times 10^{-5}$

Choose the one alternative that best completes the statement or answers the question. -Tell whether or not the number is given in scientific notation. $5.240 \times 10^{-7}$

Choose the one alternative that best completes the statement or answers the question. -Tell whether or not the number is given in scientific notation. $78.92 \times 10^{6}$

Quiz 4: Polynomials: Operations

Perform the division. - (x4−625) ÷(x+5) \left(x^{4}-625\right) \div(x+5) (x4−625) ÷(x+5)

Perform the division. - x4+3x2+10x2+1\frac{x^{4}+3 x^{2}+10}{x^{2}+1}x2+1x4+3x2+10​

Perform the division. - x8−1x4−1\frac{\mathrm{x}^{8}-1}{\mathrm{x}^{4}-1}x4−1x8−1​

Perform the division. - −16x3−4x2−6x−94x+3\frac{-16 x^{3}-4 x^{2}-6 x-9}{4 x+3}4x+3−16x3−4x2−6x−9​

Perform the division. - 12x3+3x2−35x−13−4x−5\frac{12 x^{3}+3 x^{2}-35 x-13}{-4 x-5}−4x−512x3+3x2−35x−13​

Provide an appropriate response. - (rs)4=r4 s\left(\frac{\mathrm{r}}{\mathrm{s}}\right)^{4}=\frac{\mathrm{r}^{4}}{\mathrm{~s}}(sr​)4= sr4​

Write the word or phrase that best completes each statement or answers the question. -Simplify the expressions (2x)0(2 x)^{0}(2x)0 and (2x0)\left(2 x^{0}\right)(2x0) and explain how you arrived at your answers.

Write the word or phrase that best completes each statement or answers the question. -When is the square of the sum the sum of a square; i.e., when is (A+B)2=A2+B2(A+B)^{2}=A^{2}+B^{2}(A+B)2=A2+B2 ?

Write the word or phrase that best completes each statement or answers the question. -What must be done to the dividend before beginning the "long division" process for (6x2+9x4+3x3+4x+8)÷(6x+8)?\left(6 x^{2}+9 x^{4}+3 x^{3}+4 x+8\right) \div(6 x+8) ?(6x2+9x4+3x3+4x+8)÷(6x+8)?

Choose the one alternative that best completes the statement or answers the question. -Tell whether or not the number is given in scientific notation. 5.240×10−75.240 \times 10^{-7}5.240×10−7

Choose the one alternative that best completes the statement or answers the question. -Tell whether or not the number is given in scientific notation. 78.92×10678.92 \times 10^{6}78.92×106

Perform the division. - $\left(x^{4}-625\right) \div(x+5)$

Perform the division. - $\frac{x^{4}+3 x^{2}+10}{x^{2}+1}$

Perform the division. - $\frac{\mathrm{x}^{8}-1}{\mathrm{x}^{4}-1}$

Perform the division. - $\frac{-16 x^{3}-4 x^{2}-6 x-9}{4 x+3}$

Perform the division. - $\frac{12 x^{3}+3 x^{2}-35 x-13}{-4 x-5}$

Provide an appropriate response. - $\left(\frac{\mathrm{r}}{\mathrm{s}}\right)^{4}=\frac{\mathrm{r}^{4}}{\mathrm{~s}}$

Write the word or phrase that best completes each statement or answers the question. -Simplify the expressions $(2 x)^{0}$ and $\left(2 x^{0}\right)$ and explain how you arrived at your answers.

Write the word or phrase that best completes each statement or answers the question. -When is the square of the sum the sum of a square; i.e., when is $(A+B)^{2}=A^{2}+B^{2}$ ?

Write the word or phrase that best completes each statement or answers the question. -What must be done to the dividend before beginning the "long division" process for $\left(6 x^{2}+9 x^{4}+3 x^{3}+4 x+8\right) \div(6 x+8) ?$

Choose the one alternative that best completes the statement or answers the question. -Tell whether or not the number is given in scientific notation. $5.240 \times 10^{-7}$

Choose the one alternative that best completes the statement or answers the question. -Tell whether or not the number is given in scientific notation. $78.92 \times 10^{6}$