Key Concept: Factorisation by Grouping

a) $(x + 3)(x^2 + 2)$

[Solution Description]

To factorise the expression $x^3 + 3x^2 + 2x + 6$, we can use the method of grouping. First, group the terms: $(x^3 + 3x^2) + (2x + 6)$. Factor out the common factors from each group: $x^2(x + 3) + 2(x + 3)$. Now, factor out the common binomial factor $(x + 3)$: $(x + 3)(x^2 + 2)$. Therefore, the factorised form is $(x + 3)(x^2 + 2)$.

Key Concept: Division of a monomial by another monomial

a) $-5a^2$

[Solution Description]

To simplify the expression, divide the coefficients first, then subtract the exponents of the same base: $-25a^4 \div 5a^2 = \frac{-25}{5} a^{4-2} = -5a^2$.

Key Concept: Factors of Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the expression $5xy$ can be written as $5 \times x \times y$. This is true because $5xy$ is indeed the product of 5, x, and y. The reason states that in algebraic expressions, the factors that cannot be further expressed as a product of other factors are called irreducible. This is also true because irreducible factors are those that cannot be broken down further into simpler multiplicative components. Additionally, the reason correctly explains why the factors 5, x, and y are called irreducible in the context of the assertion.

Key Concept: Division of Polynomials

b) $3y^2$

[Solution Description] Factorise the dividend $15y^4 - 30y^3$. Taking out the common factor $15y^3$, we get $15y^3(y - 2)$. Now divide this by $5y(y - 2)$. The expression becomes $\frac{15y^3(y - 2)}{5y(y - 2)} = 3y^2$ after cancelling $5y$ and $(y - 2)$.

Key Concept: Factors of Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The term $5xy$ can indeed be expressed as $5 \times x \times y$ because $5$, $x$, and $y$ are irreducible factors. This means they cannot be further broken down into simpler factors in the context of algebraic expressions. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion is true.

Key Concept: Prime Factors

d) $2^2 \times 3 \times 7$

[Solution Description]

To find the prime factor form of 84, we divide it by the smallest prime numbers until we get only prime factors.

84 divided by 2 gives 42.

42 divided by 2 gives 21.

21 divided by 3 gives 7.

7 is a prime number.

So, the prime factorization of 84 is $2 \times 2 \times 3 \times 7$, which can be written as $2^2 \times 3 \times 7$.

Key Concept: Simplification using cancellation method

a) $3a^2 + 2ab + b^2$

[Solution Description]

To simplify $\frac{15a^4b^2 + 10a^3b^3 + 5a^2b^4}{5a^2b^2}$, divide each term in the numerator by the denominator $5a^2b^2$:

$\frac{15a^4b^2}{5a^2b^2} = 3a^2$, $\frac{10a^3b^3}{5a^2b^2} = 2ab$, and $\frac{5a^2b^4}{5a^2b^2} = b^2$. Adding these results together, we get $3a^2 + 2ab + b^2$.

Key Concept: Standard Algebraic Identities

b) $(5y - 3)^2$

[Solution Description]

The given expression is $25y^2 - 30y + 9$. Observing the expression, we see that it has three terms, and the first and third terms are perfect squares: $25y^2 = (5y)^2$ and $9 = 3^2$. Additionally, the middle term $-30y$ can be expressed as $-2 \times 5y \times 3$, which fits the form of Identity II: $(a - b)^2 = a^2 - 2ab + b^2$. Therefore:

$25y^2 - 30y + 9 = (5y)^2 - 2 \times 5y \times 3 + 3^2$

Applying Identity II:

$(5y)^2 - 2 \times 5y \times 3 + 3^2 = (5y - 3)^2$

Thus, the factorisation of the expression $25y^2 - 30y + 9$ is:

$25y^2 - 30y + 9 = (5y - 3)^2$

Key Concept: Factorisation using identities

b) $(4x - 3y)^2$

[Solution Description]

First, identify if the given expression fits any of the factorisation identities. The expression has three terms: $16x^2$, $-24xy$, and $9y^2$. Notice that $16x^2 = (4x)^2$ and $9y^2 = (3y)^2$. Also, $-24xy$ can be written as $-2 \times 4x \times 3y$. This matches the form of Identity II, $(a - b)^2 = a^2 - 2ab + b^2$, where $a = 4x$ and $b = 3y$. Applying Identity II, the factorisation is:

$16x^2 - 24xy + 9y^2 = (4x - 3y)^2$

Therefore, the factorised form of the expression is $(4x - 3y)^2$.

Key Concept: Division of Algebraic Expressions, Dividing each term separately

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

To divide the polynomial $12x^3y^2 + 18x^2y^3 - 6xy^4$ by the monomial $6xy^2$, we can divide each term of the polynomial by the monomial. Let's break it down step by step:

First, consider the first term: $12x^3y^2 \div 6xy^2$.

We can write this as $\frac{12x^3y^2}{6xy^2} = \frac{12}{6} \times \frac{x^3}{x} \times \frac{y^2}{y^2} = 2x^{3-1}y^{2-2} = 2x^2$.

Next, the second term: $18x^2y^3 \div 6xy^2$.

This can be written as $\frac{18x^2y^3}{6xy^2} = \frac{18}{6} \times \frac{x^2}{x} \times \frac{y^3}{y^2} = 3x^{2-1}y^{3-2} = 3xy$.

Finally, the third term: $-6xy^4 \div 6xy^2$.

This can be written as $\frac{-6xy^4}{6xy^2} = \frac{-6}{6} \times \frac{x}{x} \times \frac{y^4}{y^2} = -y^{4-2} = -y^2$.

Combining these results, the division of the polynomial by the monomial gives $2x^2 + 3xy - y^2$.

Therefore, the Assertion is true. Now, let's evaluate the Reason.

The Reason states that division of a polynomial by a monomial requires separating the common factor from each term of the polynomial. This is also true because in the process of dividing each term by the monomial, we essentially separate the common factor (the monomial) from each term.

However, while both Assertion and Reason are true, the Reason does not directly explain why the division method works; it only describes what the process involves. Thus, the Reason is not the correct explanation of the Assertion.

Key Concept: Factorisation, Writing expressions as products of factors

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To factorise the quadratic expression $x^2 + 5x + 6$, we need to find two numbers that add up to 5 (the coefficient of the middle term) and multiply to 6 (the constant term). The numbers 2 and 3 satisfy these conditions since $2 + 3 = 5$ and $2 \times 3 = 6$. Thus, the expression can be written as $(x + 2)(x + 3)$, which is the correct factorisation. The reason provided explains the method of splitting the middle term to find factors, which is a valid and systematic approach for factorising quadratic expressions. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Factorisation of quadratic expression

b) $(x + 4)^2$

[Solution Description]

The expression $x^2 + 8x + 16$ can be recognised as a perfect square trinomial of the form $a^2 + 2ab + b^2 = (a + b)^2$. Here, $a = x$ and $b = 4$, so $x^2 + 8x + 16 = (x + 4)^2$.

Key Concept: Factorisation, Division of Algebraic Expressions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

Let's first factorise the expression $y^2 + 7y + 10$. We need to find two numbers whose product is 10 and whose sum is 7. The numbers are 5 and 2. Therefore, the factorised form is $(y + 5)(y + 2)$. Now, dividing $(y^2 + 7y + 10)$ by $(y + 5)$ is equivalent to $\frac{(y + 5)(y + 2)}{(y + 5)}$. Cancelling out $(y + 5)$ from the numerator and denominator, we get $y + 2$. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Division of a monomial by another monomial

b) $4x^2$

[Solution Description]

To solve the problem, divide the coefficients first, then subtract the exponents of the same base. Here, $12x^3 \div 3x = \frac{12}{3} x^{3-1} = 4x^2$.

Key Concept: Factorisation-based division

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] Let us first factorise the numerator $y^2 + 7y + 10$. We need to find two numbers that multiply to give 10 and add up to 7. The numbers are 5 and 2. Therefore, the factorisation is $y^2 + 7y + 10 = (y + 5)(y + 2)$. Now, dividing this by $(y + 5)$, we get $(y^2 + 7y + 10) \div (y + 5) = (y + 5)(y + 2) \div (y + 5) = y + 2$. Thus, the assertion is true. The reason is also true, and it correctly explains why the assertion is true.

Key Concept: Division of a monomial by another monomial, Division of a polynomial by a monomial

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To divide $12x^3y^2z$ by $3x^2y$, we begin by expressing both expressions in their irreducible factor forms:

$12x^3y^2z = 2 \times 2 \times 3 \times x \times x \times x \times y \times y \times z$

$3x^2y = 3 \times x \times x \times y$

Now, we cancel out the common factors:

$\frac{12x^3y^2z}{3x^2y} = \frac{2 \times 2 \times 3 \times x \times x \times x \times y \times y \times z}{3 \times x \times x \times y} = 2 \times 2 \times x \times y \times z = 4xyz$

Therefore, the assertion is true. The reason correctly explains that the division of a monomial by another monomial involves cancelling out common factors, which aligns with the procedure used to solve the assertion.

Key Concept: Division of Polynomial by Polynomial

a) $5x^2$

[Solution Description]

First, factorise the numerator and denominator. The numerator $10x^3 - 20x^2$ can be written as $10x^2(x - 2)$. The denominator $2x - 4$ can be written as $2(x - 2)$.

Now, perform the division: $\frac{10x^2(x - 2)}{2(x - 2)}$.

Cancel the common factor $(x - 2)$ to get $\frac{10x^2}{2} = 5x^2$.

Key Concept: Common factor method

c) $5p - 20$

[Solution Description]

First, factorise the numerator $5p^2 - 25p + 20$.

We take out the common factor 5: $5(p^2 - 5p + 4)$.

Next, factorise $p^2 - 5p + 4$. We look for two numbers that multiply to 4 and add to -5. These numbers are -1 and -4.

So, $p^2 - 5p + 4 = (p - 1)(p - 4)$.

Now, the expression becomes $5(p - 1)(p - 4)$.

Divide $5(p - 1)(p - 4)$ by $(p - 1)$:

$\frac{5(p - 1)(p - 4)}{(p - 1)} = 5(p - 4) = 5p - 20$.

The result of the division is $5p - 20$.

Key Concept: Exponent laws for division, Polynomial division

a) $2x^3y^2 - 3x^2y + 4x$

[Solution Description] To simplify the expression $\frac{12x^5y^3 - 18x^4y^2 + 24x^3y}{6x^2y}$, we divide each term in the numerator by the denominator separately.

First term: $\frac{12x^5y^3}{6x^2y} = 2x^{5-2}y^{3-1} = 2x^3y^2$.

Second term: $\frac{-18x^4y^2}{6x^2y} = -3x^{4-2}y^{2-1} = -3x^2y$.

Third term: $\frac{24x^3y}{6x^2y} = 4x^{3-2}y^{1-1} = 4x$.

Therefore, the simplified expression is $2x^3y^2 - 3x^2y + 4x$.

Key Concept: Prime Factor Form

b) $2 \times 5 \times 7$

[Solution Description] To find the prime factor form of 70, we start by dividing 70 by the smallest prime number, which is 2. $70 \div 2 = 35$. Next, we divide 35 by the next smallest prime number, which is 5. $35 \div 5 = 7$. Finally, we have 7, which is already a prime number. Therefore, the prime factor form of 70 is $2 \times 5 \times 7$.