GSEB Class 8 Maths Solutions Chapter 9 Algebraic Expressions and Identities Exercise 9.3

Question 1. Complete the table.
Answer: We need to find the product of the "First expression" and the "Second expression" for each row in the table.
(i) For the first row: \( a \times (b + c + d) \)
\( = a \times b + a \times c + a \times d \)
\( = ab + ac + ad \)
(ii) For the second row: \( (x + y - 5) \times 5xy \)
\( = x \times 5xy + y \times 5xy + (-5) \times 5xy \)
\( = 5x^2y + 5x^2y - 25xy \)
(iii) For the third row: \( p \times (6p^2 - 7p + 5) \)
\( = p \times 6p^2 + p \times (-7p) + p \times 5 \)
\( = 6p^3 - 7p^2 + 5p \)
(iv) For the fourth row: \( 4p^2q^2 \times (p^2 - q^2) \)
\( = 4p^2q^2 \times p^2 + 4p^2q^2 \times (-q^2) \)
\( = (4 \times 1) \times p^2q^2 \times p^2 + [4 \times (-1)] \times p^2q^2 \times q^2 \)
\( = 4p^4q^2 + (-4) \times p^2q^4 \)
\( = 4p^4q^2 - 4p^2q^4 \)
(v) For the fifth row: \( (a + b + c) \times abc \)
\( = a \times abc + b \times abc + c \times abc \)
\( = (1 \times 1) \times a \times abc + (1 \times 1) \times b \times abc + (1 \times 1) \times c \times abc \)
\( = a^2bc + ab^2c + abc^2 \)
In simple words: To complete the table, multiply the first expression by the second expression. Remember to multiply each term inside the parentheses by the term outside.

Exam Tip: When multiplying algebraic expressions, distribute each term from the first expression to every term in the second expression. Combine like terms at the end if any exist.

Question 2. Find the product?
(1) \( (a^2) \times (2a^{22}) \times (4a^{26}) \)
(2) \( (\frac{2}{3}xy) \times (-\frac{9}{10}x^2y^2) \)
(3) \( (-\frac{10}{3}pq^3) \times (\frac{6}{5}p^3q) \)
(4) \( x \times x^2 \times x^4 \times x^4 \)
Answer:
(1) To find the product of \( (a^2) \times (2a^{22}) \times (4a^{26}) \):
\( = (1 \times 2 \times 4) \times (a^2 \times a^{22} \times a^{26}) \)
\( = 8 \times a^{(2 + 22 + 26)} \)
\( = 8a^{50} \)
(2) To find the product of \( (\frac{2}{3}xy) \times (-\frac{9}{10}x^2y^2) \):
\( = (\frac{2}{3}) \times (-\frac{9}{10}) \times (xy \times x^2y^2) \)
\( = -\frac{18}{30} \times (x^{1+2}y^{1+2}) \)
\( = -\frac{3}{5}x^3y^3 \)
(3) To find the product of \( (-\frac{10}{3}pq^3) \times (\frac{6}{5}p^3q) \):
\( = (-\frac{10}{3}) \times (\frac{6}{5}) \times (pq^3 \times p^3q) \)
\( = -\frac{60}{15} \times (p^{1+3}q^{3+1}) \)
\( = -4p^4q^4 \)
(4) To find the product of \( x \times x^2 \times x^4 \times x^4 \):
\( = x^{(1+2+4+4)} \)
\( = x^{11} \)
In simple words: When multiplying terms with the same base, you add their exponents together. Multiply the numerical coefficients separately.

Exam Tip: Remember the rules of exponents: \( a^m \times a^n = a^{(m+n)} \). Also, be careful with signs, especially when multiplying negative numbers or fractions.

Question 3.
(a) Simplify \( 3x(4x - 5) + 3 \) and find its values for
(i) \( x = 3 \)
(ii) \( x = \frac{1}{2} \).
(b) Simplify \( a(a^2 + a + 1) + 5 \) and find its value for
(i) \( a = 0 \)
(ii) \( a = 1 \)
(iii) \( a = -1 \).
Answer:
(a) Simplify \( 3x(4x - 5) + 3 \):
\( 3x(4x - 5) + 3 = (3x \times 4x) + (3x \times -5) + 3 \)
\( = 12x^2 - 15x + 3 \)
(i) Now, find its value for \( x = 3 \):
Substitute \( x = 3 \) into the simplified expression \( 12x^2 - 15x + 3 \).
\( = 12(3)^2 - 15(3) + 3 \)
\( = 12(9) - 45 + 3 \)
\( = 108 - 45 + 3 \)
\( = 63 + 3 \)
\( = 66 \)
(ii) Now, find its value for \( x = \frac{1}{2} \):
Substitute \( x = \frac{1}{2} \) into the simplified expression \( 12x^2 - 15x + 3 \).
\( = 12(\frac{1}{2})^2 - 15(\frac{1}{2}) + 3 \)
\( = 12(\frac{1}{4}) - \frac{15}{2} + 3 \)
\( = 3 - \frac{15}{2} + 3 \)
\( = 6 - \frac{15}{2} \)
\( = \frac{12}{2} - \frac{15}{2} \)
\( = \frac{12 - 15}{2} \)
\( = -\frac{3}{2} \)
(b) Simplify \( a(a^2 + a + 1) + 5 \):
\( a(a^2 + a + 1) + 5 = (a \times a^2) + (a \times a) + (a \times 1) + 5 \)
\( = a^3 + a^2 + a + 5 \)
(i) Now, find its value for \( a = 0 \):
Substitute \( a = 0 \) into the simplified expression \( a^3 + a^2 + a + 5 \).
\( = (0)^3 + (0)^2 + (0) + 5 \)
\( = 0 + 0 + 0 + 5 \)
\( = 5 \)
(ii) Now, find its value for \( a = 1 \):
Substitute \( a = 1 \) into the simplified expression \( a^3 + a^2 + a + 5 \).
\( = (1)^3 + (1)^2 + (1) + 5 \)
\( = 1 + 1 + 1 + 5 \)
\( = 8 \)
(iii) Now, find its value for \( a = -1 \):
Substitute \( a = -1 \) into the simplified expression \( a^3 + a^2 + a + 5 \).
\( = (-1)^3 + (-1)^2 + (-1) + 5 \)
\( = -1 + 1 - 1 + 5 \)
\( = 4 \)
In simple words: First, use the distributive property to make the expression simpler. Then, for each part, carefully substitute the given value for the variable and calculate the result using the order of operations.

Exam Tip: When substituting negative values, especially when squaring or cubing, use parentheses to ensure correct calculation of signs. For fractions, make sure to find a common denominator before adding or subtracting.

Question 4.
(a) Add: \( p(p-q), q(q-r) \) and \( r(r-p) \)
(b) Add: \( 2x(z - x - y) \) and \( 2y(z - y - x) \)
(c) Subtract: \( 3l (l - 4m + 5n) \) from \( 4l (10n - 3m + 2l) \)
(d) Subtract: \( 3a(a + b + c) - 2b(a - b + c) \) from \( 4c(-a + b + c) \)
Answer:
(a) Add: \( p(p-q), q(q-r) \) and \( r(r-p) \)
First, expand each expression:
\( p(p-q) = p \times p - p \times q = p^2 - pq \)
\( q(q-r) = q \times q - q \times r = q^2 - qr \)
\( r(r-p) = r \times r - r \times p = r^2 - rp \)
Now, add these products:
\( (p^2 - pq) + (q^2 - qr) + (r^2 - rp) \)
\( = p^2 - pq + q^2 - qr + r^2 - rp \)
Arrange the terms:
\( = p^2 + q^2 + r^2 - pq - qr - rp \)
(b) Add: \( 2x(z - x - y) \) and \( 2y(z - y - x) \)
First, expand each expression:
\( 2x(z - x - y) = 2x \times z - 2x \times x - 2x \times y = 2xz - 2x^2 - 2xy \)
\( 2y(z - y - x) = 2y \times z - 2y \times y - 2y \times x = 2yz - 2y^2 - 2yx \)
Now, add these expressions:
\( (2xz - 2x^2 - 2xy) + (2yz - 2y^2 - 2yx) \)
Combine like terms (Note: \( -2xy \) and \( -2yx \) are like terms):
\( = 2xz - 2x^2 - 4xy + 2yz - 2y^2 \)
Rearrange for standard order:
\( = -2x^2 - 2y^2 + 2xz + 2yz - 4xy \)
(c) Subtract: \( 3l (l - 4m + 5n) \) from \( 4l (10n - 3m + 2l) \)
First, expand both expressions:
\( 3l(l - 4m + 5n) = 3l \times l - 3l \times 4m + 3l \times 5n = 3l^2 - 12lm + 15ln \)
\( 4l(10n - 3m + 2l) = 4l \times 10n - 4l \times 3m + 4l \times 2l = 40ln - 12lm + 8l^2 \)
Now, subtract the first expanded expression from the second one:
\( (40ln - 12lm + 8l^2) - (3l^2 - 12lm + 15ln) \)
\( = 40ln - 12lm + 8l^2 - 3l^2 + 12lm - 15ln \)
Group and combine like terms:
\( = (40ln - 15ln) + (-12lm + 12lm) + (8l^2 - 3l^2) \)
\( = 25ln + 0lm + 5l^2 \)
\( = 25ln + 5l^2 \)
(d) Subtract: \( 3a(a + b + c) - 2b(a - b + c) \) from \( 4c(-a + b + c) \)
First, simplify the expression to be subtracted:
\( 3a(a + b + c) - 2b(a - b + c) \)
\( = (3a^2 + 3ab + 3ac) - (2ab - 2b^2 + 2bc) \)
\( = 3a^2 + 3ab + 3ac - 2ab + 2b^2 - 2bc \)
\( = 3a^2 + ab + 3ac + 2b^2 - 2bc \)
Now, expand the expression from which we are subtracting:
\( 4c(-a + b + c) = 4c \times (-a) + 4c \times b + 4c \times c = -4ac + 4bc + 4c^2 \)
Now, subtract the first simplified expression from the second one:
\( (-4ac + 4bc + 4c^2) - (3a^2 + ab + 3ac + 2b^2 - 2bc) \)
\( = -4ac + 4bc + 4c^2 - 3a^2 - ab - 3ac - 2b^2 + 2bc \)
Group and combine like terms:
\( = (-3a^2) + (-ab) + (-2b^2) + (4c^2) + (-4ac - 3ac) + (4bc + 2bc) \)
\( = -3a^2 - ab - 2b^2 + 4c^2 - 7ac + 6bc \)
Rearrange in standard alphabetical order:
\( = -3a^2 - 2b^2 + 4c^2 - ab + 6bc - 7ac \)
In simple words: For addition or subtraction problems with algebraic expressions, first expand all products by using the distributive property. Then, identify and combine similar terms (terms with the exact same variables and exponents). When subtracting, make sure to change the sign of every term in the expression being subtracted.

Exam Tip: Pay close attention to negative signs, especially when distributing a negative number or when subtracting a multi-term expression. It's often helpful to write out all terms before combining them to reduce errors.