do now 5 - 4.4 factoring and zeros of polynomials
factor:
1. $27x^3 + 216$
2. $343x^3 - 8y^3$
factor and find all the zeros:
3. $3x^3 + 6x^2 - 45x$
4. $4x^3 + 8x^2 - 9x - 18$

Question

Accepted Answer

### Problem 1: Factor $27x^3 + 216$
# Explanation:
## Step 1: Factor out the GCF
First, we find the greatest common factor (GCF) of $27x^3$ and $216$. The GCF of $27$ and $216$ is $27$. So we factor out $27$:
$27x^3 + 216 = 27(x^3 + 8)$
## Step 2: Apply the sum of cubes formula
The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Here, $x^3 + 8 = x^3 + 2^3$, so $a = x$ and $b = 2$. Applying the formula:
$x^3 + 2^3 = (x + 2)(x^2 - 2x + 4)$
## Step 3: Combine the factors
Substituting back into the expression from Step 1, we get:
$27(x + 2)(x^2 - 2x + 4)$
# Answer:
$27(x + 2)(x^2 - 2x + 4)$

### Problem 2: Factor $343x^3 - 8y^3$
# Explanation:
## Step 1: Identify the difference of cubes
The expression $343x^3 - 8y^3$ is a difference of cubes. We know that $343x^3 = (7x)^3$ and $8y^3 = (2y)^3$. The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Here, $a = 7x$ and $b = 2y$.
## Step 2: Apply the difference of cubes formula
Using the formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ with $a = 7x$ and $b = 2y$:
$(7x)^3 - (2y)^3 = (7x - 2y)((7x)^2 + (7x)(2y) + (2y)^2)$
## Step 3: Simplify the second factor
Simplify $(7x)^2 + (7x)(2y) + (2y)^2$:
$(7x)^2 = 49x^2$, $(7x)(2y) = 14xy$, $(2y)^2 = 4y^2$. So the second factor is $49x^2 + 14xy + 4y^2$.
# Answer:
$(7x - 2y)(49x^2 + 14xy + 4y^2)$

### Problem 3: Factor and find all the zeros of $3x^3 + 6x^2 - 45x$
# Explanation:
## Step 1: Factor out the GCF
First, factor out the greatest common factor (GCF) of $3x^3$, $6x^2$, and $-45x$. The GCF is $3x$:
$3x^3 + 6x^2 - 45x = 3x(x^2 + 2x - 15)$
## Step 2: Factor the quadratic
Factor the quadratic $x^2 + 2x - 15$. We need two numbers that multiply to $-15$ and add to $2$. The numbers are $5$ and $-3$. So:
$x^2 + 2x - 15 = (x + 5)(x - 3)$
## Step 3: Combine the factors
Putting it all together, the factored form is:
$3x(x + 5)(x - 3)$
## Step 4: Find the zeros
To find the zeros, set the factored form equal to zero:
$3x(x + 5)(x - 3) = 0$
Using the zero - product property ($ab = 0$ implies $a = 0$ or $b = 0$), we have:
$3x = 0$ or $x + 5 = 0$ or $x - 3 = 0$
Solving for $x$:
- For $3x = 0$, we get $x = 0$.
- For $x + 5 = 0$, we get $x=-5$.
- For $x - 3 = 0$, we get $x = 3$.
# Answer:
Factored form: $3x(x + 5)(x - 3)$; Zeros: $x = 0$, $x=-5$, $x = 3$

### Problem 4: Factor and find all the zeros of $4x^3 + 8x^2 - 9x - 18$
# Explanation:
## Step 1: Group the terms
Group the first two terms and the last two terms:
$(4x^3 + 8x^2)+(-9x - 18)$
## Step 2: Factor out the GCF from each group
Factor out $4x^2$ from the first group and $-9$ from the second group:
$4x^2(x + 2)-9(x + 2)$
## Step 3: Factor out the common binomial factor
Now, we can factor out the common binomial factor $(x + 2)$:
$(x + 2)(4x^2 - 9)$
## Step 4: Factor the difference of squares
The expression $4x^2 - 9$ is a difference of squares, since $4x^2=(2x)^2$ and $9 = 3^2$. The difference of squares formula is $a^2 - b^2=(a + b)(a - b)$. Here, $a = 2x$ and $b = 3$, so:
$4x^2 - 9=(2x + 3)(2x - 3)$
## Step 5: Combine the factors
The fully factored form is:
$(x + 2)(2x + 3)(2x - 3)$
## Step 6: Find the zeros
Set the factored form equal to zero:
$(x + 2)(2x + 3)(2x - 3)=0$
Using the zero - product property:
- $x + 2 = 0$ gives $x=-2$.
- $2x+3 = 0$ gives $x =-\frac{3}{2}$.
- $2x - 3 = 0$ gives $x=\frac{3}{2}$.
# Answer:
Factored form: $(x + 2)(2x + 3)(2x - 3)$; Zeros: $x=-2$, $x =-\frac{3}{2}$, $x=\frac{3}{2}$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Factor \(27x^3 + 216\)

Step 1: Factor out the GCF

Step 2: Apply the sum of cubes formula

Step 3: Combine the factors

Step 1: Identify the difference of cubes

Step 2: Apply the difference of cubes formula

Step 3: Simplify the second factor

Step 1: Factor out the GCF

Step 2: Factor the quadratic

Step 3: Combine the factors

Step 4: Find the zeros

Answer:

Problem 2: Factor \(343x^3 - 8y^3\)