EXERCISE 6 (a) Factorise: $$ \begin{array}{lll} ext { (1) } 6 x^{3}+12 x & ext { (2) } 6 x^{3}+4 x^{2} & ext { (3) } 5 x^{3}+5 x \ ext { (4) } 12 x^{3}-18 x^{2} & ext { (5) } & 3 x^{2}-9 y+12 x y\end{array} $$ $$ \begin{array}{ll} ext { (7) } 16 m^{4} n^{8}-8 m^{3} n^{7}+36 m^{2} n^{3} & \ ext { (7) } & \ ext { (6 } b^{2}-64 a b\end{array} $$

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EXERCISE 6 (a) Factorise: $$ \begin{array}{lll}	ext { (1) } 6 x^{3}+12 x & 	ext { (2) } 6 x^{3}+4 x^{2} & 	ext { (3) } 5 x^{3}+5 x \ 	ext { (4) } 12 x^{3}-18 x^{2} & 	ext { (5) } & 3 x^{2}-9 y+12 x y\end{array} $$ $$ \begin{array}{ll}	ext { (7) } 16 m^{4} n^{8}-8 m^{3} n^{7}+36 m^{2} n^{3} & \ 	ext { (7) } & \ 	ext { (6 } b^{2}-64 a b\end{array} $$

UpStudy AI · Accepted Answer

Sure! Let's factorize each of the given expressions step by step.

### (a) Factorise:

#### (1) $$6x^{3} + 12x$$

**Step 1:** Identify the Greatest Common Factor (GCF).

- **Coefficients:** The GCF of 6 and 12 is **6**.
- **Variables:** The smallest power of $$x$$ present is $$x$$.

**Step 2:** Factor out the GCF.

$$
6x^{3} + 12x = 6x(x^{2} + 2)
$$

**Answer:** $$6x(x^{2} + 2)$$

---

#### (2) $$6x^{3} + 4x^{2}$$

**Step 1:** Identify the GCF.

- **Coefficients:** The GCF of 6 and 4 is **2**.
- **Variables:** The smallest power of $$x$$ present is $$x^{2}$$.

**Step 2:** Factor out the GCF.

$$
6x^{3} + 4x^{2} = 2x^{2}(3x + 2)
$$

**Answer:** $$2x^{2}(3x + 2)$$

---

#### (3) $$5x^{3} + 5x$$

**Step 1:** Identify the GCF.

- **Coefficients:** The GCF of 5 and 5 is **5**.
- **Variables:** The smallest power of $$x$$ present is $$x$$.

**Step 2:** Factor out the GCF.

$$
5x^{3} + 5x = 5x(x^{2} + 1)
$$

**Answer:** $$5x(x^{2} + 1)$$

---

#### (4) $$12x^{3} - 18x^{2}$$

**Step 1:** Identify the GCF.

- **Coefficients:** The GCF of 12 and 18 is **6**.
- **Variables:** The smallest power of $$x$$ present is $$x^{2}$$.

**Step 2:** Factor out the GCF.

$$
12x^{3} - 18x^{2} = 6x^{2}(2x - 3)
$$

**Answer:** $$6x^{2}(2x - 3)$$

---

#### (5) $$3x^{2} - 9y + 12xy$$

**Step 1:** Identify the GCF for the entire expression.

- The terms $$3x^{2}$$, $$-9y$$, and $$12xy$$ all share a common factor of **3**.

**Step 2:** Factor out the GCF.

$$
3x^{2} - 9y + 12xy = 3(x^{2} + 4xy - 3y)
$$

**Step 3 (Optional):** Check if the quadratic can be factored further.

- The expression inside the parentheses, $$x^{2} + 4xy - 3y$$, does not factor neatly with integer coefficients.

**Answer:** $$3(x^{2} + 4xy - 3y)$$

---

#### (6) $$b^{2} - 64ab$$

**Step 1:** Identify the GCF.

- **Variables:** Both terms contain at least one $$b$$.

**Step 2:** Factor out the GCF.

$$
b^{2} - 64ab = b(b - 64a)
$$

**Answer:** $$b(b - 64a)$$

---

#### (7) $$16m^{4}n^{8} - 8m^{3}n^{7} + 36m^{2}n^{3}$$

**Step 1:** Identify the GCF.

- **Coefficients:** The GCF of 16, -8, and 36 is **4**.
- **Variables:** The smallest powers are $$m^{2}$$ and $$n^{3}$$.

**Step 2:** Factor out the GCF.

$$
16m^{4}n^{8} - 8m^{3}n^{7} + 36m^{2}n^{3} = 4m^{2}n^{3}(4m^{2}n^{5} - 2mn^{4} + 9)
$$

**Step 3 (Optional):** Check if the remaining polynomial can be factored further.

- The expression $$4m^{2}n^{5} - 2mn^{4} + 9$$ does not factor neatly with integer coefficients.

**Answer:** $$4m^{2}n^{3}(4m^{2}n^{5} - 2mn^{4} + 9)$$

---

### Summary of Factorizations:

1. **$$6x^{3} + 12x = 6x(x^{2} + 2)$$**
2. **$$6x^{3} + 4x^{2} = 2x^{2}(3x + 2)$$**
3. **$$5x^{3} + 5x = 5x(x^{2} + 1)$$**
4. **$$12x^{3} - 18x^{2} = 6x^{2}(2x - 3)$$**
5. **$$3x^{2} - 9y + 12xy = 3(x^{2} + 4xy - 3y)$$**
6. **$$b^{2} - 64ab = b(b - 64a)$$**
7. **$$16m^{4}n^{8} - 8m^{3}n^{7} + 36m^{2}n^{3} = 4m^{2}n^{3}(4m^{2}n^{5} - 2mn^{4} + 9)$$**

If you need further assistance or explanations for any of these factorizations, feel free to ask!
Here are the factorizations: 1. $$6x^{3} + 12x = 6x(x^{2} + 2)$$ 2. $$6x^{3} + 4x^{2} = 2x^{2}(3x + 2)$$ 3. $$5x^{3} + 5x = 5x(x^{2} + 1)$$ 4. $$12x^{3} - 18x^{2} = 6x^{2}(2x - 3)$$ 5. $$3x^{2} - 9y + 12xy = 3(x^{2} + 4xy - 3y)$$ 6. $$b^{2} - 64ab = b(b - 64a)$$ 7. $$16m^{4}n^{8} - 8m^{3}n^{7} + 36m^{2}n^{3} = 4m^{2}n^{3}(4m^{2}n^{5} - 2mn^{4} + 9)$$

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