Question 11:

What is the greatest common factor of the terms of the polynomial $-16y^4 + 12y^2 - 4y$?

Step 1

Identify the coefficients and variables in each term:

• $-16y^4$-$12y^2$-$-4y$

Step 2

Find the greatest common factor (GCF) of the coefficients:

• Coefficients: $-16$,$12$,$-4$- GCF of$-16$,$12$, and$-4$is$4$.

Step 3

Find the lowest power of $y$ present in all terms:

• $y^4$,$y^2$,$y$- The lowest power is$y$.

Final Answer

The greatest common factor is $4y$.

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Question 12:

The combined area of two rectangular rugs is $84 \, \text{m}^2$. If each of the dimensions is a prime number, what are the four dimensions?

Step 1

List the prime factors of $84$:

• $84 = 2 \times 2 \times 3 \times 7$

Step 2

Check the options for combinations of these prime factors:

• A. $2, 3, 3, 7$- B.$3, 3, 4, 7$- C.$2, 2, 3, 3$- D.$2, 2, 3, 7$

Step 3

Calculate the area for each option:

• A. $2 \times 3 \times 3 \times 7 = 126$ (Incorrect)
• B. $3 \times 3 \times 4 \times 7 = 252$ (Incorrect)
• C. $2 \times 2 \times 3 \times 3 = 36$ (Incorrect)
• D. $2 \times 2 \times 3 \times 7 = 84$ (Correct)

Final Answer

D. $2, 2, 3, 7$

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Question 13:

Factor out the greatest common factor from the terms of the polynomial $2x^3 - 5x^2 + 25$.

Step 1

Identify the terms:

• $2x^3$-$-5x^2$-$25$

Step 2

Find the GCF of the coefficients:

• Coefficients: $2$,$-5$,$25$- The GCF is$1$ (since there is no common factor other than 1).

Step 3

Check if any variable can be factored out:

• The smallest power of $x$is$x^2$in the first two terms, but$25$has no$x$.

Step 4

Choose the correct option:

• A. $x^2(2x - 5) + 25$ (Incorrect)
• B. Already fully factored (Correct)
• C. $5x(x^2 - x + 5)$ (Incorrect)
• D. $2x^3 - 5(x^2 - 5)$ (Incorrect)

Final Answer

B. Already fully factored

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