Question 1

Factor the perfect square trinomial. 25s² + 20s + 4

Accepted Answer

The correct factorization of the given perfect square trinomial $25s^2 + 20s + 4$ is $(5s + 2)^2$, which matches option D. This is because the square root of $25s^2$ is $5s$, the square root of $4$ is $2$, and twice their product ($2 \times 5s \times 2$) gives the middle term $20s$.

Question 2

Factor by grouping. z³ + 4z² + 8z + 32

Accepted Answer

By grouping the first two terms, we get z³ + 4z² = z²(z + 4). Similarly, the last two terms can be grouped as 8z + 32 = 8(z + 4). Therefore, the entire expression can be written as (z² + 8)(z + 4) which is option C.

Question 3

Factor the sum or difference of cubes. 27n³ + 8

Accepted Answer

The sum of cubes can be factored as $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Here, $27n^3 + 8$ can be rewritten as $(3n)^3 + 2^3$, which factors into $(3n + 2)((3n)^2 - 3n*2 + 2^2)$ or $(3n + 2)(9n^2 - 6n + 4)$.

Question 4

Factor the difference of two squares. 25t² - 4

Accepted Answer

The difference of squares formula is a^2 - b^2 = (a + b)(a - b). Here, 25t^2 - 4 can be rewritten as (5t)^2 - 2^2, so it factors to (5t + 2)(5t - 2).

Question 5

Find the area of the shaded region in the figure.Write your result as a polynomial in standard form.

Accepted Answer

The shaded region can be divided into a trapezoid and two triangles. The area of the trapezoid is: (1/2) x (6x + 2x) x 1 = 4x The area of each triangle is: (1/2) x 2x x 1 = x So, the total area of the shaded region is: 4x + 2x = 6x Writing the answer in standard form, we get: 6x² - x, which corresponds to choice A.

Question 6

Factor out the common factor. 30q - 5

Accepted Answer

The common factor between 30q and -5 is 5, so when you factor it out, you get 5(6q - 1).

Question 7

Factor the sum or difference of cubes. r³ - 125

Accepted Answer

The expression can be factored using the difference of cubes formula, $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, where $a = r$ and $b = 5$.

Question 8

The number of committees consisting of exactly 3 people that can be formed from a group of n people is given by the polynomial  Find the number of committees consisting of exactly 3 people that can be formed from a group of  people. Round to the nearest whole number.

Accepted Answer

The answer of The number of committees consisting of exactly...

Question 9

Find the product below.

Accepted Answer

The answer of Find the product below. 
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Question 10

Factor the perfect square trinomial.

Accepted Answer

The answer of Factor the perfect square trinomial. 
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Question 11

Factor out the common factor. 5r(2r + 3) + (2r + 3)

Accepted Answer

The common factor is (2r + 3), so we can factor it out of both terms to get (2r + 3)(5r + 1). Option D is the only choice that gives this factorization.

Question 12

Factor the trinomial. u² - 8u + 12

Accepted Answer

First, factor out the greatest common factor, which in this case is (u - 2):
uS1U1P12S1S1P0 - 8u + 12 = (u - 2)(uS1U1P12S1S1 + 6u - 6)
Now, the remaining trinomial can be factored using the AC method:
(uS1U1P12S1S1 + 6u - 6) = (u + 2)(u - 3)
Putting both factors together, we get:
(u - 2)(u + 2)(u - 3)
However, the original question asks for a factorization in which all parentheses have no numerical coefficient other than 1. So, we can rearrange the factors as:
(u + 2)(u - 2)(u - 3)
which is equivalent to choice D.

Question 13

After 2 years, an investment of $550 compounded annually at an interest rate r will yield an amount of 550(1 + r)². Write this polynomial in standard form.

Accepted Answer

Using the formula for compound interest, we have:

Final amount = Principal * (1 + rate)^time

Plugging in the given values, we get:

Final amount = 550 * (1 + r)^2

Expanding the square, we get:

Final amount = 550 * (1 + 2r + r^2)

Multiplying through by the coefficient, we get:

Final amount = 550r^2 + 1100r + 550

So the polynomial in standard form is:

550r^2 + 1100r + 550

which matches choice D.

Question 14

Factor the trinomial. 9m² + 9m + 2

Accepted Answer

The trinomial 9m^2 + 9m + 2 can be factored as (3m + 2)(3m + 1) because when expanded, it results in the original trinomial: 3m * 3m = 9m^2, 3m * 1 + 2 * 3m = 9m, and 2 * 1 = 2.

Question 15

Completely factor the expression. 4u² - 64

Accepted Answer

Firstly, focus on the terms with the variable "u". Factor out "4u" from the first two terms and "4" from the last term. This gives us:
4u(S1U1P1)2(S1S1P0)
Now, factor out "4" from the remaining terms:
4(u + 4)(S1U1P1)2(S1S1P0)
Finally, simplify the exponents of the first term:
4(u + 4)(S1U1P12)(S1)(P0)
The expression is now completely factored. Option D matches this result: 4(u + 4)(u - 4).

Question 16

On an expressway, the recommended safe distance between cars in feet is given by  where v is the speed of the car in miles per hour.Find the safe distance when  miles per hour.Round your answer to the nearest tenth.

Accepted Answer

The answer of On an expressway, the recommended safe distance...

Question 17

Factor the difference of two squares. 49 - (m + 4) ²

Accepted Answer

(m + 4)² - (m + 11)² = (m + 4 + m + 11)(m + 4 - m - 11) = (2m + 15)(-7) = (m + 11)(3 - m)

Question 18

Completely factor the expression. 8z³ - 46z² + 56z

Accepted Answer

The correct factorization of the given expression $8z^3 - 46z^2 + 56z$ is found by first factoring out the greatest common factor, which is $2z$, resulting in $2z(4z^2 - 23z + 28)$. Further factoring the quadratic expression gives $2z(z - 4)(4z - 7)$.

Question 19

Factor by grouping. 15w⁷ + 25w⁴ + 6w³ + 10

Accepted Answer

We can group the first two terms and the last two terms: (15w⁷ + 25w⁴) + (6w³ + 10) = 5w⁴(3w³ + 5) + 2(3w³ + 5). Now we have a common factor of (3w³ + 5), which we can factor out: (3w³ + 5)(5w⁴ + 2). Therefore, the answer is D.

Question 20

Factor out the common factor. 35n⁴ + 5n³ + 45n²

Accepted Answer

The common factor among all terms is $5n^{12}$, and factoring it out correctly results in option B, which simplifies the expression accurately by extracting the common factor and adjusting the powers of $n$ accordingly.

Quiz 1: Fundamental Concepts of Algebra

Factor the perfect square trinomial. 25s² + 20s + 4

Factor by grouping. z³ + 4z² + 8z + 32

Factor the sum or difference of cubes. 27n³ + 8

Factor the difference of two squares. 25t² - 4

Find the area of the shaded region in the figure.Write your result as a polynomial in standard form.

Factor out the common factor. 30q - 5

Factor the sum or difference of cubes. r³ - 125

The number of committees consisting of exactly 3 people that can be formed from a group of n people is given by the polynomial Find the number of committees consisting of exactly 3 people that can be formed from a group of people. Round to the nearest whole number.

Find the product below.

Factor the perfect square trinomial.

Factor out the common factor. 5r(2r + 3) + (2r + 3)

Factor the trinomial. u² - 8u + 12

After 2 years, an investment of $550 compounded annually at an interest rate r will yield an amount of 550(1 + r) ². Write this polynomial in standard form.

Factor the trinomial. 9m² + 9m + 2

Completely factor the expression. 4u² - 64

On an expressway, the recommended safe distance between cars in feet is given by where v is the speed of the car in miles per hour.Find the safe distance when miles per hour.Round your answer to the nearest tenth.

Factor the difference of two squares. 49 - (m + 4) ²

Completely factor the expression. 8z³ - 46z² + 56z

Factor by grouping. 15w⁷ + 25w⁴ + 6w³ + 10

Factor out the common factor. 35n⁴ + 5n³ + 45n²

Quiz 1: Fundamental Concepts of Algebra

Factor the perfect square trinomial. 25s2 + 20s + 4

Factor by grouping. z3 + 4z2 + 8z + 32

Factor the sum or difference of cubes. 27n3 + 8

Factor the difference of two squares. 25t2 - 4

Find the area of the shaded region in the figure.Write your result as a polynomial in standard form.

Factor out the common factor. 30q - 5

Factor the sum or difference of cubes. r3 - 125

The number of committees consisting of exactly 3 people that can be formed from a group of n people is given by the polynomial Find the number of committees consisting of exactly 3 people that can be formed from a group of people. Round to the nearest whole number.

Find the product below.

Factor the perfect square trinomial.

Factor out the common factor. 5r(2r + 3) + (2r + 3)

Factor the trinomial. u2 - 8u + 12

After 2 years, an investment of $550 compounded annually at an interest rate r will yield an amount of 550(1 + r) 2. Write this polynomial in standard form.

Factor the trinomial. 9m2 + 9m + 2

Completely factor the expression. 4u2 - 64

On an expressway, the recommended safe distance between cars in feet is given by where v is the speed of the car in miles per hour.Find the safe distance when miles per hour.Round your answer to the nearest tenth.

Factor the difference of two squares. 49 - (m + 4) 2

Completely factor the expression. 8z3 - 46z2 + 56z

Factor by grouping. 15w7 + 25w4 + 6w3 + 10

Factor out the common factor. 35n4 + 5n3 + 45n2

Factor the perfect square trinomial. 25s² + 20s + 4

Factor by grouping. z³ + 4z² + 8z + 32

Factor the sum or difference of cubes. 27n³ + 8

Factor the difference of two squares. 25t² - 4

Factor the sum or difference of cubes. r³ - 125

Factor the trinomial. u² - 8u + 12

After 2 years, an investment of $550 compounded annually at an interest rate r will yield an amount of 550(1 + r) ². Write this polynomial in standard form.

Factor the trinomial. 9m² + 9m + 2

Completely factor the expression. 4u² - 64

Factor the difference of two squares. 49 - (m + 4) ²

Completely factor the expression. 8z³ - 46z² + 56z

Factor by grouping. 15w⁷ + 25w⁴ + 6w³ + 10

Factor out the common factor. 35n⁴ + 5n³ + 45n²