16. $27z^3 - 9z$
17. $x^2 - 6x + 9$
18. $15p^3 - 3p - 54$
19. $x^2 - 9$
20. $8x^2 - 40x + 50$
21. $-4x^2 - 10x + 6$
mixed practice b. factor each completely. box your answers!
22. $9y^2 - 36$
23. $16x^2 - 80x + 100$
24. $-x^2 + 18x - 81$
25. $64x^2 - 16$
26. $\frac{1}{2}y^2 - \frac{1}{2}$
27. $30n^2b - 87nb + 30b$
28. $2f^2 - 16f$
29. $12a^2 - 75$
30. $m^4 - 1$

Question

Accepted Answer

Let's solve problem 16: $27z^3 - 9z$ using factoring.

# Explanation:
## Step 1: Find the GCF
The greatest common factor (GCF) of $27z^3$ and $9z$ is $9z$.
$27z^3 - 9z = 9z(3z^2 - 1)$

# Answer:
$\boxed{9z(3z^2 - 1)}$

---

Let's solve problem 17: $x^2 - 6x + 9$ using factoring.

# Explanation:
## Step 1: Recognize the perfect square trinomial
The expression $x^2 - 6x + 9$ is a perfect square trinomial of the form $a^2 - 2ab + b^2 = (a - b)^2$. Here, $a = x$ and $b = 3$ (since $2ab = 2 \cdot x \cdot 3 = 6x$).
$x^2 - 6x + 9 = (x - 3)^2$

# Answer:
$\boxed{(x - 3)^2}$

---

Let's solve problem 18: $15p^2 - 3p - 54$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $15p^2$, $-3p$, and $-54$ is $3$.
$15p^2 - 3p - 54 = 3(5p^2 - p - 18)$

## Step 2: Factor the quadratic
Factor $5p^2 - p - 18$. We need two numbers that multiply to $5 \cdot (-18) = -90$ and add to $-1$. These numbers are $-10$ and $9$.
$5p^2 - p - 18 = 5p^2 + 9p - 10p - 18$
$= p(5p + 9) - 2(5p + 9)$
$= (5p + 9)(p - 2)$

## Step 3: Combine the factors
$15p^2 - 3p - 54 = 3(5p + 9)(p - 2)$

# Answer:
$\boxed{3(5p + 9)(p - 2)}$

---

Let's solve problem 19: $x^2 - 9$ using factoring.

# Explanation:
## Step 1: Recognize the difference of squares
The expression $x^2 - 9$ is a difference of squares of the form $a^2 - b^2 = (a - b)(a + b)$. Here, $a = x$ and $b = 3$ (since $9 = 3^2$).
$x^2 - 9 = (x - 3)(x + 3)$

# Answer:
$\boxed{(x - 3)(x + 3)}$

---

Let's solve problem 20: $8x^2 - 40x + 50$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $8x^2$, $-40x$, and $50$ is $2$.
$8x^2 - 40x + 50 = 2(4x^2 - 20x + 25)$

## Step 2: Recognize the perfect square trinomial
The expression $4x^2 - 20x + 25$ is a perfect square trinomial of the form $a^2 - 2ab + b^2 = (a - b)^2$. Here, $a = 2x$ and $b = 5$ (since $2ab = 2 \cdot 2x \cdot 5 = 20x$).
$4x^2 - 20x + 25 = (2x - 5)^2$

## Step 3: Combine the factors
$8x^2 - 40x + 50 = 2(2x - 5)^2$

# Answer:
$\boxed{2(2x - 5)^2}$

---

Let's solve problem 21: $-4x^2 - 10x + 6$ using factoring.

# Explanation:
## Step 1: Factor out the negative GCF
The GCF of $-4x^2$, $-10x$, and $6$ is $-2$.
$-4x^2 - 10x + 6 = -2(2x^2 + 5x - 3)$

## Step 2: Factor the quadratic
Factor $2x^2 + 5x - 3$. We need two numbers that multiply to $2 \cdot (-3) = -6$ and add to $5$. These numbers are $6$ and $-1$.
$2x^2 + 5x - 3 = 2x^2 + 6x - x - 3$
$= 2x(x + 3) - 1(x + 3)$
$= (2x - 1)(x + 3)$

## Step 3: Combine the factors
$-4x^2 - 10x + 6 = -2(2x - 1)(x + 3)$

# Answer:
$\boxed{-2(2x - 1)(x + 3)}$

---

Let's solve problem 22: $9y^2 - 36$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $9y^2$ and $-36$ is $9$.
$9y^2 - 36 = 9(y^2 - 4)$

## Step 2: Recognize the difference of squares
The expression $y^2 - 4$ is a difference of squares of the form $a^2 - b^2 = (a - b)(a + b)$. Here, $a = y$ and $b = 2$ (since $4 = 2^2$).
$y^2 - 4 = (y - 2)(y + 2)$

## Step 3: Combine the factors
$9y^2 - 36 = 9(y - 2)(y + 2)$

# Answer:
$\boxed{9(y - 2)(y + 2)}$

---

Let's solve problem 23: $16x^2 - 80x + 100$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $16x^2$, $-80x$, and $100$ is $4$.
$16x^2 - 80x + 100 = 4(4x^2 - 20x + 25)$

## Step 2: Recognize the perfect square trinomial
The expression $4x^2 - 20x + 25$ is a perfect square trinomial of the form $a^2 - 2ab + b^2 = (a - b)^2$. Here, $a = 2x$ and $b = 5$ (since $2ab = 2 \cdot 2x \cdot 5 = 20x$).
$4x^2 - 20x + 25 = (2x - 5)^2$

## Step 3: Combine the factors
$16x^2 - 80x + 100 = 4(2x - 5)^2$

# Answer:
$\boxed{4(2x - 5)^2}$

---

Let's solve problem 24: $-x^2 + 18x - 81$ using factoring.

# Explanation:
## Step 1: Factor out the negative GCF
The GCF of $-x^2$, $18x$, and $-81$ is $-1$.
$-x^2 + 18x - 81 = -1(x^2 - 18x + 81)$

## Step 2: Recognize the perfect square trinomial
The expression $x^2 - 18x + 81$ is a perfect square trinomial of the form $a^2 - 2ab + b^2 = (a - b)^2$. Here, $a = x$ and $b = 9$ (since $2ab = 2 \cdot x \cdot 9 = 18x$).
$x^2 - 18x + 81 = (x - 9)^2$

## Step 3: Combine the factors
$-x^2 + 18x - 81 = -1(x - 9)^2 = -(x - 9)^2$

# Answer:
$\boxed{-(x - 9)^2}$

---

Let's solve problem 25: $64x^2 - 16$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $64x^2$ and $-16$ is $16$.
$64x^2 - 16 = 16(4x^2 - 1)$

## Step 2: Recognize the difference of squares
The expression $4x^2 - 1$ is a difference of squares of the form $a^2 - b^2 = (a - b)(a + b)$. Here, $a = 2x$ and $b = 1$ (since $1 = 1^2$).
$4x^2 - 1 = (2x - 1)(2x + 1)$

## Step 3: Combine the factors
$64x^2 - 16 = 16(2x - 1)(2x + 1)$

# Answer:
$\boxed{16(2x - 1)(2x + 1)}$

---

Let's solve problem 26: $\frac{1}{2}y^2 - \frac{1}{2}$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $\frac{1}{2}y^2$ and $-\frac{1}{2}$ is $\frac{1}{2}$.
$\frac{1}{2}y^2 - \frac{1}{2} = \frac{1}{2}(y^2 - 1)$

## Step 2: Recognize the difference of squares
The expression $y^2 - 1$ is a difference of squares of the form $a^2 - b^2 = (a - b)(a + b)$. Here, $a = y$ and $b = 1$ (since $1 = 1^2$).
$y^2 - 1 = (y - 1)(y + 1)$

## Step 3: Combine the factors
$\frac{1}{2}y^2 - \frac{1}{2} = \frac{1}{2}(y - 1)(y + 1)$

# Answer:
$\boxed{\frac{1}{2}(y - 1)(y + 1)}$

---

Let's solve problem 27: $30n^2b - 87nb + 30b$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $30n^2b$, $-87nb$, and $30b$ is $3b$.
$30n^2b - 87nb + 30b = 3b(10n^2 - 29n + 10)$

## Step 2: Factor the quadratic
Factor $10n^2 - 29n + 10$. We need two numbers that multiply to $10 \cdot 10 = 100$ and add to $-29$. These numbers are $-25$ and $-4$.
$10n^2 - 29n + 10 = 10n^2 - 25n - 4n + 10$
$= 5n(2n - 5) - 2(2n - 5)$
$= (2n - 5)(5n - 2)$

## Step 3: Combine the factors
$30n^2b - 87nb + 30b = 3b(2n - 5)(5n - 2)$

# Answer:
$\boxed{3b(2n - 5)(5n - 2)}$

---

Let's solve problem 28: $2f^2 - 16f$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $2f^2$ and $-16f$ is $2f$.
$2f^2 - 16f = 2f(f - 8)$

# Answer:
$\boxed{2f(f - 8)}$

---

Let's solve problem 29: $12a^2 - 75$ using factoring.

# Explanation:
## Step 1: Find the GCF
The GCF of $12a^2$ and $-75$ is $3$.
$12a^2 - 75 = 3(4a^2 - 25)$

## Step 2: Recognize the difference of squares
The expression $4a^2 - 25$ is a difference of squares of the form $a^2 - b^2 = (a - b)(a + b)$. Here, $a = 2a$ and $b = 5$ (since $25 = 5^2$).
$4a^2 - 25 = (2a - 5)(2a + 5)$

## Step 3: Combine the factors
$12a^2 - 75 = 3(2a - 5)(2a + 5)$

# Answer:
$\boxed{3(2a - 5)(2a + 5)}$

---

Let's solve problem 30: $m^4 - 1$ using factoring.

# Explanation:
## Step 1: Recognize the difference of squares
The expression $m^4 - 1$ is a difference of squares of the form $a^2 - b^2 = (a - b)(a + b)$. Here, $a = m^2$ and $b = 1$ (since $1 = 1^2$).
$m^4 - 1 = (m^2 - 1)(m^2 + 1)$

## Step 2: Factor the difference of squares again
The expression $m^2 - 1$ is also a difference of squares: $m^2 - 1 = (m - 1)(m + 1)$

## Step 3: Combine the factors
$m^4 - 1 = (m - 1)(m + 1)(m^2 + 1)$

# Answer:
$\boxed{(m - 1)(m + 1)(m^2 + 1)}$

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QUESTION IMAGE

Question

Explanation:

Step 1: Find the GCF

Step 1: Recognize the perfect square trinomial

Step 1: Find the GCF

Step 2: Factor the quadratic

Step 3: Combine the factors

Answer: