GSEB Class 8 Maths Solutions Chapter 9 Algebraic Expressions and Identities Exercise 9.4

Question 1. Multiply the binomials.
(i) \( (2x + 5) \) and \( (4x - 3) \)
(ii) \( (y - 8) \) and \( (3y - 4) \)
(iii) \( (2.5l - 0.5m) \) and \( (2.5l + 0.5m) \)
(iv) \( (a + 3b) \) and \( (x + 5) \)
(v) \( (2pq + 3q^2) \) and \( (3pq - 2q^2) \)
(vi) \( (\frac{3}{4}a^2 + 3b^2) \) and \( 4(a^2 - \frac{2}{3}b^2) \)
Answer:
(i) To multiply \( (2x + 5) \) by \( (4x - 3) \), we distribute each term from the first binomial to the second. First, distribute \( 2x \) to \( (4x - 3) \) and then distribute \( 5 \) to \( (4x - 3) \). This gives us \( 2x(4x - 3) + 5(4x - 3) \). Next, we multiply the terms: \( (2x \times 4x) - (2x \times 3) + (5 \times 4x) - (5 \times 3) \). This simplifies to \( 8x^2 - 6x + 20x - 15 \). Finally, we combine the like terms \( -6x \) and \( 20x \) to get \( 14x \). So, the ultimate product is \( 8x^2 + 14x - 15 \).
(ii) To find the product of \( (y - 8) \) and \( (3y - 4) \), we distribute \( y \) and \( -8 \) to the second binomial. This gives \( y(3y - 4) - 8(3y - 4) \). Expanding these terms, we get \( (3y \times y) - (4 \times y) - (8 \times 3y) - (8 \times -4) \). This simplifies to \( 3y^2 - 4y - 24y + 32 \). Combining the like terms \( -4y \) and \( -24y \), the final product is \( 3y^2 - 28y + 32 \).
(iii) We multiply the binomials \( (2.5l - 0.5m) \) and \( (2.5l + 0.5m) \) by first distributing \( 2.5l \) to \( (2.5l + 0.5m) \) and then \( -0.5m \) to \( (2.5l + 0.5m) \). This results in \( 2.5l(2.5l + 0.5m) - 0.5m(2.5l + 0.5m) \). Expanding each part, we get \( (2.5l \times 2.5l) + (2.5l \times 0.5m) - (0.5m \times 2.5l) - (0.5m \times 0.5m) \). This becomes \( 6.25l^2 + 1.25lm - 1.25lm - 0.25m^2 \). The middle terms \( 1.25lm \) and \( -1.25lm \) cancel each other, giving \( 0lm \). So, the simplified product is \( 6.25l^2 - 0.25m^2 \).
(iv) To find the product of \( (a + 3b) \) and \( (x + 5) \), we multiply each term in the first binomial by each term in the second. We distribute \( a \) to \( (x + 5) \) and \( 3b \) to \( (x + 5) \). This results in \( a(x + 5) + 3b(x + 5) \). Expanding these, we get \( (a \times x) + (a \times 5) + (3b \times x) + (3b \times 5) \). This simplifies to \( ax + 5a + 3bx + 15b \). Since there are no like terms, this is the final expanded form.
(v) We expand the expression \( (2pq + 3q^2) \times (3pq - 2q^2) \) by distributing \( 2pq \) to \( (3pq - 2q^2) \) and \( 3q^2 \) to \( (3pq - 2q^2) \). This gives \( 2pq(3pq - 2q^2) + 3q^2(3pq - 2q^2) \). Performing the multiplications, we get \( (2pq \times 3pq) - (2pq \times 2q^2) + (3q^2 \times 3pq) - (3q^2 \times 2q^2) \). This simplifies to \( 6p^2q^2 - 4pq^3 + 9pq^3 - 6q^4 \). Combining the like terms \( -4pq^3 \) and \( 9pq^3 \) gives \( 5pq^3 \). So, the final product is \( 6p^2q^2 + 5pq^3 - 6q^4 \).
(vi) We first distribute \( \frac{3}{4}a^2 \) to the second term and \( 3b^2 \) to the second term. The expression becomes \( \frac{3}{4}a^2 \times 4(a^2 - \frac{2}{3}b^2) + 3b^2 \times 4(a^2 - \frac{2}{3}b^2) \). Simplifying this, we get \( 3a^2(a^2 - \frac{2}{3}b^2) + 12b^2(a^2 - \frac{2}{3}b^2) \). Expanding each part, we perform the multiplications: \( (3a^2 \times a^2) - (3a^2 \times \frac{2}{3}b^2) + (12b^2 \times a^2) - (12b^2 \times \frac{2}{3}b^2) \). This simplifies to \( 3a^4 - 2a^2b^2 + 12a^2b^2 - 8b^4 \). Combining the like terms \( -2a^2b^2 \) and \( 12a^2b^2 \), we get \( 10a^2b^2 \). So, the final product is \( 3a^4 + 10a^2b^2 - 8b^4 \).
In simple words: To multiply two binomials, multiply each term from the first one by each term from the second one. Then, add all the results and combine any similar terms together to simplify your answer.

Exam Tip: Remember to apply the distributive property correctly and carefully combine like terms to avoid errors.

Question 2. Find the product:
1. \( (5 - 2x)(3 + x) \)
2. \( (x + 7y)(7x - y) \)
Answer:
1. To find the product of \( (5 - 2x) \) and \( (3 + x) \), we multiply each term of the first binomial by each term of the second. Distribute \( 5 \) to \( (3 + x) \) and \( -2x \) to \( (3 + x) \). This gives \( 5(3 + x) - 2x(3 + x) \). Expanding these terms, we get \( (5 \times 3) + (5 \times x) - (2x \times 3) - (2x \times x) \). This simplifies to \( 15 + 5x - 6x - 2x^2 \). Combining the like terms \( 5x \) and \( -6x \) gives \( -x \). The final product is \( 15 - x - 2x^2 \).
2. We expand the expression \( (x + 7y)(7x - y) \) by distributing \( x \) to \( (7x - y) \) and \( 7y \) to \( (7x - y) \). This becomes \( x(7x - y) + 7y(7x - y) \). Multiplying each term, we get \( (x \times 7x) - (x \times y) + (7y \times 7x) - (7y \times y) \). This simplifies to \( 7x^2 - xy + 49xy - 7y^2 \). Combining the like terms \( -xy \) and \( 49xy \), we get \( 48xy \). So, the final product is \( 7x^2 + 48xy - 7y^2 \).
In simple words: Just like with the previous question, multiply every part of the first expression by every part of the second expression. Then, bring together any similar parts to make the answer as short as possible.

Exam Tip: Pay close attention to the signs (+ or -) when multiplying terms, especially when a negative sign is involved, as this is a common source of mistakes.

Question 3. Simplify:
1. \( (x^2 - 5)(x + 5) + 25 \)
2. \( (a^2 + 5)(b^3 + 3) + 5 \)
3. \( (t + s^2)(t^2 - s) \)
4. \( (a + b)(c - d) + (a - b)(c + d) + 2(ac + bd) \)
5. \( (x + y)(2x + y) + (x + 2y)(x - y) \)
6. \( (x + y)(x^2 - xy + y^2) \)
7. \( (1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y \)
8. \( (a + b + c)(a + b - c) \)
Answer:
1. We begin by multiplying the first two binomials \( (x^2 - 5)(x + 5) \). Distribute \( x^2 \) to \( (x + 5) \) and \( -5 \) to \( (x + 5) \). This gives \( x^2(x + 5) - 5(x + 5) \). Expanding, we get \( (x^2 \times x) + (x^2 \times 5) - (5 \times x) - (5 \times 5) \). This simplifies to \( x^3 + 5x^2 - 5x - 25 \). Then, we add the final term \( +25 \). The \( -25 \) and \( +25 \) cancel each other out. So, the simplified expression is \( x^3 + 5x^2 - 5x \).
2. To simplify \( (a^2 + 5)(b^3 + 3) + 5 \), we first multiply the two binomials. Distribute \( a^2 \) to \( (b^3 + 3) \) and \( 5 \) to \( (b^3 + 3) \). This results in \( a^2(b^3 + 3) + 5(b^3 + 3) \). Expanding these multiplications, we get \( (a^2 \times b^3) + (a^2 \times 3) + (5 \times b^3) + (5 \times 3) \). This simplifies to \( a^2b^3 + 3a^2 + 5b^3 + 15 \). Finally, we add the remaining \( +5 \) to the expression. Combining the constant terms \( 15 + 5 \) gives \( 20 \). So, the simplified form is \( a^2b^3 + 3a^2 + 5b^3 + 20 \).
3. We expand the expression \( (t + s^2)(t^2 - s) \) by distributing \( t \) to \( (t^2 - s) \) and \( s^2 \) to \( (t^2 - s) \). This yields \( t(t^2 - s) + s^2(t^2 - s) \). Multiplying each term, we get \( (t \times t^2) - (t \times s) + (s^2 \times t^2) + (s^2 \times -s) \). This simplifies to \( t^3 - ts + s^2t^2 - s^3 \). Since there are no like terms, this is the final simplified expression.
4. We expand each product separately for \( (a + b)(c - d) + (a - b)(c + d) + 2(ac + bd) \).
For \( (a + b)(c - d) \), we get \( ac - ad + bc - bd \).
For \( (a - b)(c + d) \), we get \( ac + ad - bc - bd \).
For \( 2(ac + bd) \), we get \( 2ac + 2bd \).
Now, we combine all these expanded parts: \( (ac - ad + bc - bd) + (ac + ad - bc - bd) + (2ac + 2bd) \).
Group the like terms:
\( (ac + ac + 2ac) + (-ad + ad) + (bc - bc) + (-bd - bd + 2bd) \).
This simplifies to \( 4ac + 0 + 0 + 0 \).
Therefore, the final simplified expression is \( 4ac \).
5. We expand each product for \( (x + y)(2x + y) + (x + 2y)(x - y) \).
For \( (x + y)(2x + y) \): Distribute \( x \) and \( y \). This gives \( x(2x + y) + y(2x + y) = 2x^2 + xy + 2xy + y^2 = 2x^2 + 3xy + y^2 \).
For \( (x + 2y)(x - y) \): Distribute \( x \) and \( 2y \). This gives \( x(x - y) + 2y(x - y) = x^2 - xy + 2xy - 2y^2 = x^2 + xy - 2y^2 \).
Now, we add the two simplified expressions:
\( (2x^2 + 3xy + y^2) + (x^2 + xy - 2y^2) \).
Combine the like terms:
\( (2x^2 + x^2) + (3xy + xy) + (y^2 - 2y^2) \).
This simplifies to \( 3x^2 + 4xy - y^2 \).
6. We expand the product \( (x + y)(x^2 - xy + y^2) \) by distributing \( x \) to \( (x^2 - xy + y^2) \) and \( y \) to \( (x^2 - xy + y^2) \). This gives \( x(x^2 - xy + y^2) + y(x^2 - xy + y^2) \). Multiplying each term, we get \( (x \times x^2) - (x \times xy) + (x \times y^2) + (y \times x^2) - (y \times xy) + (y \times y^2) \). This simplifies to \( x^3 - x^2y + xy^2 + yx^2 - xy^2 + y^3 \). Now, we combine like terms: \( -x^2y \) and \( +yx^2 \) (which is \( +x^2y \)) cancel out. Also, \( +xy^2 \) and \( -xy^2 \) cancel out. So, the simplified expression is \( x^3 + y^3 \).
7. First, we multiply the two polynomials: \( (1.5x - 4y)(1.5x + 4y + 3) \).
We distribute each term from the first polynomial to the second:
\( 1.5x(1.5x + 4y + 3) - 4y(1.5x + 4y + 3) \)
Expanding these, we get:
\( (1.5x \times 1.5x) + (1.5x \times 4y) + (1.5x \times 3) - (4y \times 1.5x) - (4y \times 4y) - (4y \times 3) \)
This simplifies to:
\( 2.25x^2 + 6xy + 4.5x - 6xy - 16y^2 - 12y \)
Now, we include the remaining terms from the original expression: \( - 4.5x + 12y \).
So, the full expression is:
\( 2.25x^2 + 6xy + 4.5x - 6xy - 16y^2 - 12y - 4.5x + 12y \)
Combine like terms:
\( (2.25x^2) + (6xy - 6xy) + (4.5x - 4.5x) + (-16y^2) + (-12y + 12y) \)
This simplifies to \( 2.25x^2 + 0 + 0 - 16y^2 + 0 \).
Therefore, the simplified expression is \( 2.25x^2 - 16y^2 \).
8. We can simplify the expression \( (a + b + c)(a + b - c) \) using the difference of squares identity, \( (X+Y)(X-Y) = X^2 - Y^2 \).
Here, let \( X = (a + b) \) and \( Y = c \).
So, \( (a + b + c)(a + b - c) = ((a + b) + c)((a + b) - c) \).
Applying the identity, we get \( (a + b)^2 - c^2 \).
Now, expand \( (a + b)^2 \) using the identity \( (A+B)^2 = A^2 + 2AB + B^2 \).
So, \( (a + b)^2 = a^2 + 2ab + b^2 \).
Substituting this back, the expression becomes \( a^2 + 2ab + b^2 - c^2 \).
This can also be written as \( a^2 + b^2 - c^2 + 2ab \).
In simple words: To simplify expressions with multiple terms, first expand any multiplications by distributing terms. Then, group and combine all similar terms (like all \( x^2 \) terms, all \( xy \) terms, or all numbers) to get the shortest possible answer.

Exam Tip: For complex algebraic expressions, break down the problem into smaller steps. First, perform all multiplications, then combine like terms, and look for algebraic identities that can simplify the process.