EXERCISE 12 Fully factorise each of the following expressions. 1. $$ p-q-x(q-p) $$ 2. $$ x(a+b)-(b+a) $$ 3. $$ a x-b x+a y-b y-a+b $$ 4. $$ 3 x-6 x y-2 y+1 $$ 5. $$ 4 p^{3}-2 p-p q^{2}+q $$ 6. $$ 6 m^{2}-2 m n-3 m p+n p $$ 7. $$ 8 a m-10 a n-12 b m+15 b m $$ 8. $$ 3 x^{2}-2 x y+2 y^{2}-3 x y $$ 9. $$ x^{3}-x^{2} y+y^{3}-x y^{2} $$ 10. $$ m^{2}-9 n^{2}-3 k m+9 k n

Question

EXERCISE 12 Fully factorise each of the following expressions. 1. $$ p-q-x(q-p) $$ 2. $$ x(a+b)-(b+a) $$ 3. $$ a x-b x+a y-b y-a+b $$ 4. $$ 3 x-6 x y-2 y+1 $$ 5. $$ 4 p^{3}-2 p-p q^{2}+q $$ 6. $$ 6 m^{2}-2 m n-3 m p+n p $$ 7. $$ 8 a m-10 a n-12 b m+15 b m $$ 8. $$ 3 x^{2}-2 x y+2 y^{2}-3 x y $$ 9. $$ x^{3}-x^{2} y+y^{3}-x y^{2} $$ 10. $$ m^{2}-9 n^{2}-3 k m+9 k n< $$

UpStudy AI · Accepted Answer

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

---

### **Problem 1**
**Expression:** $$ p - q - x(q - p) $$

**Solution:**
$$
\begin{align*}
p - q - x(q - p) &= p - q - xq + xp \
&= p - q + xp - xq \
&= (1 + x)p - (1 + x)q \
&= (p - q)(1 + x) \
&= (p - q)(x + 1)
\end{align*}
$$

**Factorized Form:** $$(p - q)(x + 1)$$

---

### **Problem 2**
**Expression:** $$ x(a + b) - (b + a) $$

**Solution:**
$$
\begin{align*}
x(a + b) - (a + b) &= (x - 1)(a + b) \
&= (x - 1)(a + b) \
\end{align*}
$$

**Factorized Form:** $$(x - 1)(a + b)$$

---

### **Problem 3**
**Expression:** $$ ax - bx + ay - by - a + b $$

**Solution:**
$$
\begin{align*}
ax - bx + ay - by - a + b &= x(a - b) + y(a - b) - (a - b) \
&= (a - b)(x + y - 1) \
\end{align*}
$$

**Factorized Form:** $$(a - b)(x + y - 1)$$

---

### **Problem 4**
**Expression:** $$ 3x - 6xy - 2y + 1 $$

**Solution:**
$$
\begin{align*}
3x - 6xy - 2y + 1 &= 3x(1 - 2y) - 2y + 1 \
&= 3x(1 - 2y) - (2y - 1) \
&= 3x(1 - 2y) + (1 - 2y) \
&= (1 - 2y)(3x + 1) \
\end{align*}
$$

**Factorized Form:** $$(1 - 2y)(3x + 1)$$

---

### **Problem 5**
**Expression:** $$ 4p^{3} - 2p - pq^{2} + q $$

**Solution:**
$$
\begin{align*}
4p^{3} - 2p - pq^{2} + q &= p(4p^{2} - 2 - q^{2}) + q \
&= p(4p^{2} - q^{2} - 2) + q \
\text{Notice that } 4p^{2} - q^{2} &= (2p - q)(2p + q) \
\text{But } 4p^{2} - q^{2} - 2 \text{ doesn't factor nicely, so factor by grouping:} \
4p^{3} - pq^{2} - 2p + q &= p(4p^{2} - q^{2} - 2) + q \
\end{align*}
$$
It appears that the expression doesn't factor nicely over integers. However, we can factor out a common term:
$$
= p(4p^{2} - q^{2} - 2) + q = \text{No further factorization possible.}
$$

**Factorized Form:** $$ p(4p^{2} - q^{2} - 2) + q $$

---

### **Problem 6**
**Expression:** $$ 6m^{2} - 2mn - 3mp + np $$

**Solution:**
$$
\begin{align*}
6m^{2} - 2mn - 3mp + np &= 6m^{2} - 3mp - 2mn + np \
&= 3m(2m - p) - n(2m - p) \
&= (2m - p)(3m - n) \
\end{align*}
$$

**Factorized Form:** $$(2m - p)(3m - n)$$

---

### **Problem 7**
**Expression:** $$ 8am - 10an - 12bm + 15bn $$

**Solution:**
$$
\begin{align*}
8am - 10an - 12bm + 15bn &= 2a(4m - 5n) - 3b(4m - 5n) \
&= (4m - 5n)(2a - 3b) \
\end{align*}
$$

**Factorized Form:** $$(4m - 5n)(2a - 3b)$$

---

### **Problem 8**
**Expression:** $$ 3x^{2} - 2xy + 2y^{2} - 3xy $$

**Solution:**
$$
\begin{align*}
3x^{2} - 2xy + 2y^{2} - 3xy &= 3x^{2} - 5xy + 2y^{2} \
&= (3x^{2} - 3xy) - 2xy + 2y^{2} \
&= 3x(x - y) - 2y(x - y) \
&= (x - y)(3x - 2y) \
\end{align*}
$$

**Factorized Form:** $$(x - y)(3x - 2y)$$

---

### **Problem 9**
**Expression:** $$ x^{3} - x^{2}y + y^{3} - xy^{2} $$

**Solution:**
$$
\begin{align*}
x^{3} - x^{2}y + y^{3} - xy^{2} &= x^{2}(x - y) + y^{2}(y - x) \
&= x^{2}(x - y) - y^{2}(x - y) \
&= (x - y)(x^{2} - y^{2}) \
&= (x - y)(x - y)(x + y) \
&= (x - y)^2 (x + y) \
\end{align*}
$$

**Factorized Form:** $$(x - y)^2 (x + y)$$

---

### **Problem 10**
**Expression:** $$ m^{2} - 9n^{2} - 3km + 9kn $$

**Solution:**
$$
\begin{align*}
m^{2} - 9n^{2} - 3km + 9kn &= m^{2} - 3km - 9n^{2} + 9kn \
&= m(m - 3k) - 9n(n - k) \
&= m(m - 3k) + 9n(k - n) \
&= m(m - 3k) - 9n(n - k) \
\text{Notice that } m^2 - 3km &= m(m - 3k) \
& \text{and } -9n^{2} + 9kn = 9n(k - n) \
&= (m + 3n)(m - 3k - 3n) \
\end{align*}
$$
Upon closer inspection, another approach using grouping:
$$
\begin{align*}
m^{2} - 3km - 9n^{2} + 9kn &= m(m - 3k) - 9n(n - k) \
&= m(m - 3k) + 9n(k - n) \
&= m(m - 3k) + 9n(k - n) \
\text{Factor by grouping:} \
&= (m + 3n)(m - 3k - 3n) \
\end{align*}
$$

**Factorized Form:** $$(m + 3n)(m - 3k - 3n)$$

---

**Note:** If you encounter the inequality sign "<" at the end of Problem 10, it appears to be a typo. The factorization above assumes it's an equation.
1. $$ p - q - x(q - p) = (p - q)(x + 1) $$ 2. $$ x(a + b) - (b + a) = (x - 1)(a + b) $$ 3. $$ ax - bx + ay - by - a + b = (a - b)(x + y - 1) $$ 4. $$ 3x - 6xy - 2y + 1 = (1 - 2y)(3x + 1) $$ 5. $$ 4p^{3} - 2p - pq^{2} + q = p(4p^{2} - q^{2} - 2) + q $$ 6. $$ 6m^{2} - 2mn - 3mp + np = (2m - p)(3m - n) $$ 7. $$ 8am - 10an - 12bm + 15bn = (4m - 5n)(2a - 3b) $$ 8. $$ 3x^{2} - 2xy + 2y^{2} - 3xy = (x - y)(3x - 2y) $$ 9. $$ x^{3} - x^{2}y + y^{3} - xy^{2} = (x - y)^2 (x + y) $$ 10. $$ m^{2} - 9n^{2} - 3km + 9kn = (m + 3n)(m - 3k - 3n) $$

Solución de tutoría real

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions