factor each by gcf. (3 points each)
1) $5x^2 + 30$ 2) $4x^2 - 7x$ 3) $12x^3 + 6x^2$

factor each difference of squares. (3 points each)
4) $25a^4 - 81$ 5) $100x^2y^6 - 1$

factor each perfect square trinomial. (3 points each)
6) $x^2 - 10x + 25$ 7) $9x^2 + 12xy + 4y^2$

factor each using the x - method (multiply/add method). (4 points each)
8) $x^2 - 27x + 50$ 9) $x^2 - 6x - 16$

Question

factor each by gcf. (3 points each)
1) $5x^2 + 30$ 2) $4x^2 - 7x$ 3) $12x^3 + 6x^2$

factor each difference of squares. (3 points each)
4) $25a^4 - 81$ 5) $100x^2y^6 - 1$

factor each perfect square trinomial. (3 points each)
6) $x^2 - 10x + 25$ 7) $9x^2 + 12xy + 4y^2$

factor each using the x - method (multiply/add method). (4 points each)
8) $x^2 - 27x + 50$ 9) $x^2 - 6x - 16$

Accepted Answer

### 1) Factoring $5x^2 + 30$ by GCF
# Explanation:
## Step 1: Find GCF of coefficients
The coefficients are 5 and 30. The GCF of 5 and 30 is 5.
## Step 2: Factor out GCF
Factor out 5 from $5x^2 + 30$: $5(x^2 + 6)$
# Answer: $5(x^2 + 6)$

### 2) Factoring $4x^2 - 7x$ by GCF
# Explanation:
## Step 1: Find GCF of terms
The terms are $4x^2$ and $-7x$. The GCF of $4x^2$ and $-7x$ is $x$.
## Step 2: Factor out GCF
Factor out $x$ from $4x^2 - 7x$: $x(4x - 7)$
# Answer: $x(4x - 7)$

### 3) Factoring $12x^3 + 6x^2$ by GCF
# Explanation:
## Step 1: Find GCF of coefficients and variables
Coefficients: GCF of 12 and 6 is 6. Variables: GCF of $x^3$ and $x^2$ is $x^2$. So overall GCF is $6x^2$.
## Step 2: Factor out GCF
Factor out $6x^2$ from $12x^3 + 6x^2$: $6x^2(2x + 1)$
# Answer: $6x^2(2x + 1)$

### 4) Factoring $25a^4 - 81$ (Difference of Squares)
# Explanation:
## Step 1: Identify as difference of squares
$25a^4=(5a^2)^2$ and $81 = 9^2$. So it is $(5a^2)^2-9^2$.
## Step 2: Apply difference of squares formula $a^2 - b^2=(a + b)(a - b)$
Using the formula with $a = 5a^2$ and $b = 9$: $(5a^2 + 9)(5a^2 - 9)$. Note that $5a^2 - 9$ is also a difference of squares ($(\sqrt{5}a)^2-3^2$), but usually we factor over integers, so we stop here.
# Answer: $(5a^2 + 9)(5a^2 - 9)$

### 5) Factoring $100x^2y^6 - 1$ (Difference of Squares)
# Explanation:
## Step 1: Identify as difference of squares
$100x^2y^6=(10xy^3)^2$ and $1 = 1^2$. So it is $(10xy^3)^2-1^2$.
## Step 2: Apply difference of squares formula $a^2 - b^2=(a + b)(a - b)$
Using the formula with $a = 10xy^3$ and $b = 1$: $(10xy^3 + 1)(10xy^3 - 1)$
# Answer: $(10xy^3 + 1)(10xy^3 - 1)$

### 6) Factoring $x^2 - 10x + 25$ (Perfect Square Trinomial)
# Explanation:
## Step 1: Identify as perfect square trinomial
A perfect square trinomial is of the form $a^2-2ab + b^2=(a - b)^2$. Here, $x^2=x^2$, $-10x=-2\times x\times5$, and $25 = 5^2$. So $a=x$, $b = 5$.
## Step 2: Apply the formula
Using $a^2-2ab + b^2=(a - b)^2$ with $a=x$ and $b = 5$: $(x - 5)^2$
# Answer: $(x - 5)^2$

### 7) Factoring $9x^2 + 12xy + 4y^2$ (Perfect Square Trinomial)
# Explanation:
## Step 1: Identify as perfect square trinomial
A perfect square trinomial is of the form $a^2+2ab + b^2=(a + b)^2$. Here, $9x^2=(3x)^2$, $12xy=2\times3x\times2y$, and $4y^2=(2y)^2$. So $a = 3x$, $b=2y$.
## Step 2: Apply the formula
Using $a^2+2ab + b^2=(a + b)^2$ with $a = 3x$ and $b = 2y$: $(3x + 2y)^2$
# Answer: $(3x + 2y)^2$

### 8) Factoring $x^2 - 27x + 50$ using X - Method
# Explanation:
## Step 1: Identify $a$, $b$, $c$
For $ax^2+bx + c$ (here $a = 1$, $b=-27$, $c = 50$). We need two numbers that multiply to $a\times c=1\times50 = 50$ and add up to $b=-27$.
## Step 2: Find the two numbers
The numbers are $-25$ and $-2$ since $(-25)\times(-2)=50$ and $-25+(-2)=-27$.
## Step 3: Rewrite the middle term and factor by grouping
Rewrite $-27x$ as $-25x-2x$: $x^2-25x-2x + 50$. Group as $(x^2-25x)+(-2x + 50)$. Factor out $x$ from first group and $-2$ from second group: $x(x - 25)-2(x - 25)$. Then factor out $(x - 25)$: $(x - 25)(x - 2)$
# Answer: $(x - 25)(x - 2)$

### 9) Factoring $x^2 - 6x - 16$ using X - Method
# Explanation:
## Step 1: Identify $a$, $b$, $c$
For $ax^2+bx + c$ (here $a = 1$, $b=-6$, $c=-16$). We need two numbers that multiply to $a\times c=1\times(-16)=-16$ and add up to $b=-6$.
## Step 2: Find the two numbers
The numbers are $-8$ and $2$ since $(-8)\times2=-16$ and $-8 + 2=-6$.
## Step 3: Rewrite the middle term and factor by grouping
Rewrite $-6x$ as $-8x+2x$: $x^2-8x + 2x-16$. Group as $(x^2-8x)+(2x - 16)$. Factor out $x$ from first group and $2$ from second group: $x(x - 8)+2(x - 8)$. Then factor out $(x - 8)$: $(x - 8)(x + 2)$
# Answer: $(x - 8)(x + 2)$

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QUESTION IMAGE

Question

Explanation:

1) Factoring \(5x^2 + 30\) by GCF

Step 1: Find GCF of coefficients

Step 2: Factor out GCF

Step 1: Find GCF of terms

Step 2: Factor out GCF

Step 1: Find GCF of coefficients and variables

Step 2: Factor out GCF

Answer:

2) Factoring \(4x^2 - 7x\) by GCF