k) $$ 5 x^{2}+15 x+10 $$ 1) $$ 3 x^{2}-18 x^{2}+24 x $$ m) $$ 2 x^{4}-12 x^{3}+10 x^{2} $$ n) $$ x^{3}-10 x^{2}+21 x $$ 0) $$ 5 x^{3} y+25 x^{2} y+30 x y $$

Question

UpStudy AI · Accepted Answer

Sure, let's factor each of the given polynomial expressions step by step.

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### **k) $$ 5x^{2} + 15x + 10 $$**

**Step 1:** Factor out the greatest common factor (GCF), which is **5**.

$$
5x^{2} + 15x + 10 = 5(x^{2} + 3x + 2)
$$

**Step 2:** Factor the quadratic expression $$ x^{2} + 3x + 2 $$.

$$
x^{2} + 3x + 2 = (x + 1)(x + 2)
$$

**Final Factored Form:**

$$
5(x + 1)(x + 2)
$$

---

### **1) $$ 3x^{3} - 18x^{2} + 24x $$**

*Note: It seems there might be a typo in the original expression. Assuming it's $$ 3x^{3} - 18x^{2} + 24x $$.*

**Step 1:** Factor out the GCF, which is **3x**.

$$
3x^{3} - 18x^{2} + 24x = 3x(x^{2} - 6x + 8)
$$

**Step 2:** Factor the quadratic expression $$ x^{2} - 6x + 8 $$.

$$
x^{2} - 6x + 8 = (x - 2)(x - 4)
$$

**Final Factored Form:**

$$
3x(x - 2)(x - 4)
$$

---

### **m) $$ 2x^{4} - 12x^{3} + 10x^{2} $$**

**Step 1:** Factor out the GCF, which is **2x²**.

$$
2x^{4} - 12x^{3} + 10x^{2} = 2x^{2}(x^{2} - 6x + 5)
$$

**Step 2:** Factor the quadratic expression $$ x^{2} - 6x + 5 $$.

$$
x^{2} - 6x + 5 = (x - 1)(x - 5)
$$

**Final Factored Form:**

$$
2x^{2}(x - 1)(x - 5)
$$

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### **n) $$ x^{3} - 10x^{2} + 21x $$**

**Step 1:** Factor out the GCF, which is **x**.

$$
x^{3} - 10x^{2} + 21x = x(x^{2} - 10x + 21)
$$

**Step 2:** Factor the quadratic expression $$ x^{2} - 10x + 21 $$.

$$
x^{2} - 10x + 21 = (x - 3)(x - 7)
$$

**Final Factored Form:**

$$
x(x - 3)(x - 7)
$$

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### **0) $$ 5x^{3}y + 25x^{2}y + 30xy $$**

**Step 1:** Factor out the GCF, which is **5xy**.

$$
5x^{3}y + 25x^{2}y + 30xy = 5xy(x^{2} + 5x + 6)
$$

**Step 2:** Factor the quadratic expression $$ x^{2} + 5x + 6 $$.

$$
x^{2} + 5x + 6 = (x + 2)(x + 3)
$$

**Final Factored Form:**

$$
5xy(x + 2)(x + 3)
$$

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Feel free to reach out if you have any more questions or need further assistance!
Here are the factored forms of the given polynomials: - **k)** $$ 5(x + 1)(x + 2) $$ - **1)** $$ 3x(x - 2)(x - 4) $$ - **m)** $$ 2x^{2}(x - 1)(x - 5) $$ - **n)** $$ x(x - 3)(x - 7) $$ - **0)** $$ 5xy(x + 2)(x + 3) $$

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