NCERT Solutions for Class 8 Maths Chapter 8: Algebraic Expressions and Identities - Exercise 8.4

1. Multiply the following binomials.

i. $\left( {2x + 5} \right)$ and $\left( {4x - 3} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {2x + 5} \right) \times \left( {4x - 3} \right) = 2x \times \left( {4x - 3} \right) + 5 \times \left( {4x - 3} \right)$

$\left( {2x + 5} \right) \times \left( {4x - 3} \right) = 8{x^2} - 6x + 20x - 15$

On adding the like terms, we get,

$\left( {2x + 5} \right) \times \left( {4x - 3} \right) = 8{x^2} + 14x - 15$

ii. $\left( {y - 8} \right)$ and $\left( {3y - 4} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {y - 8} \right) \times \left( {3y - 4} \right) = y \times \left( {3y - 4} \right) - 8 \times \left( {3y - 4} \right)$

$\left( {y - 8} \right) \times \left( {3y - 4} \right) = 3{y^2} - 4y - 24y + 32$

On adding the like terms, we get,

$\left( {y - 8} \right) \times \left( {3y - 4} \right) = 3{y^2} - 28y + 32$

iii. $\left( {2.5l - 0.5m} \right)$ and $\left( {2.5l + 0.5m} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {2.5l - 0.5m} \right) \times \left( {2.5l + 0.5m} \right) = 2.5l \times \left( {2.5l + 0.5m} \right) - 0.5m \times \left( {2.5l + 0.5m} \right)$

$\left( {2.5l - 0.5m} \right) \times \left( {2.5l + 0.5m} \right) = 6.25{l^2} + 1.251m - 1.25ml - 0.25{m^2}$

On adding the like terms, we get,

$\left( {2.5l - 0.5m} \right) \times \left( {2.5l + 0.5m} \right) = 6.25{l^2} - 0.25{m^2}$

iv. $\left( {a + 3b} \right)$ and $\left( {x + 5} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {a + 3b} \right) \times \left( {x + 5} \right) = a \times \left( {x + 5} \right) + 3b \times \left( {x + 5} \right)$

$\left( {a + 3b} \right) \times \left( {x + 5} \right) = ax + 5a + 3bx + 15b$

v. $\left( {2pq + 3{q^2}} \right)$ and $\left( {3pq - 2{q^2}} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

\[\left( {2pq + 3{q^2}} \right) \times \left( {3pq - 2{q^2}} \right) = 2pq \times \left( {3pq - 2{q^2}} \right) + 3{q^2} \times \left( {3pq - 2{q^2}} \right)\]

\[\left( {2pq + 3{q^2}} \right) \times \left( {3pq - 2{q^2}} \right) = 6{p^2}{q^2} - 4p{q^3} + 9p{q^3} - 6{q^4}\]

On adding the like terms, we get,

\[\left( {2pq + 3{q^2}} \right) \times \left( {3pq - 2{q^2}} \right) = 6{p^2}{q^2} + 5p{q^3} - 6{q^4}\]

vi. $\left( {\dfrac{3}{4}{a^2} + 3{b^2}} \right)$ and $4\left( {{a^2} - \dfrac{2}{3}{b^2}} \right)$

Ans: On multiplying the bracket of the second term and simplifying it, we get,

$4\left( {{a^2} - \dfrac{2}{3}{b^2}} \right) = 4{a^2} - \dfrac{8}{3}{b^2}$

We will multiply the given terms by applying the distributive property and then simplify the equation further. 

\[\left( {\dfrac{3}{4}{a^2} + 3{b^2}} \right) \times \left( {4{a^2} - \dfrac{8}{3}{b^2}} \right) = \dfrac{3}{4}{a^2} \times \left( {4{a^2} - \dfrac{8}{3}{b^2}} \right) + 3{b^2} \times \left( {4{a^2} - \dfrac{8}{3}{b^2}} \right)\]

\[\left( {\dfrac{3}{4}{a^2} + 3{b^2}} \right) \times \left( {4{a^2} - \dfrac{8}{3}{b^2}} \right) = 3{a^4} - \dfrac{{24}}{{12}}{a^2}{b^2} + 12{b^2}{a^2} - 8{b^4}\]

\[\left( {\dfrac{3}{4}{a^2} + 3{b^2}} \right) \times \left( {4{a^2} - \dfrac{8}{3}{b^2}} \right) = 3{a^4} - 2{a^2}{b^2} + 12{b^2}{a^2} - 8{b^4}\]

On adding the like terms, we get,

\[\left( {\dfrac{3}{4}{a^2} + 3{b^2}} \right) \times \left( {4{a^2} - \dfrac{8}{3}{b^2}} \right) = 3{a^4} + 10{a^2}{b^2} - 8{b^4}\]

2. Find the product of the following binomials.

i. $\left( {5 - 2x} \right)\left( {3 + x} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {5 - 2x} \right)\left( {3 + x} \right) = 5\left( {3 + x} \right) - 2x\left( {3 + x} \right)$

$\left( {5 - 2x} \right)\left( {3 + x} \right) = 15 + 5x - 6x - 2{x^2}$

On adding the like terms, we get,

$\left( {5 - 2x} \right)\left( {3 + x} \right) = 15 - x - 2{x^2}$

ii. $\left( {x + 7y} \right)\left( {7x - y} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {x + 7y} \right)\left( {7x - y} \right) = x\left( {7x - y} \right) + 7y\left( {7x - y} \right)$

$\left( {x + 7y} \right)\left( {7x - y} \right) = 7{x^2} - xy + 49xy - 7{y^2}$

On adding the like terms, we get,

$\left( {x + 7y} \right)\left( {7x - y} \right) = 7{x^2} + 48xy - 7{y^2}$

iii. $\left( {{a^2} + b} \right)\left( {a + {b^2}} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {{a^2} + b} \right)\left( {a + {b^2}} \right) = {a^2}\left( {a + {b^2}} \right) + b\left( {a + {b^2}} \right)$

$\left( {{a^2} + b} \right)\left( {a + {b^2}} \right) = {a^3} + {a^2}{b^2} + ab + {b^3}$

iv. $\left( {{p^2} - {q^2}} \right)\left( {2p + q} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {{p^2} - {q^2}} \right)\left( {2p + q} \right) = {p^2}\left( {2p + q} \right) - {q^2}\left( {2p + q} \right)$

$\left( {{p^2} - {q^2}} \right)\left( {2p + q} \right) = 2{p^3} + {p^2}q - 2{q^2}p - {q^3}$

3. Simplify the following expressions. 

i. $\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25$

Ans: We will first multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = {x^2}\left( {x + 5} \right) - 5\left( {x + 5} \right) + 25$

$\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = {x^3} + {5x^2} - 5x - 25 + 25$

On adding the like terms, we get,

$\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = {x^3} + {5x^2} - 5x$

ii. $\left( {{a^2} + 5} \right)\left( {{b^3} + 3} \right) + 5$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {{a^2} + 5} \right)\left( {{b^3} + 3} \right) + 5 = {a^2}\left( {{b^3} + 3} \right) + 5\left( {{b^3} + 3} \right) + 5$

$\left( {{a^2} + 5} \right)\left( {{b^3} + 3} \right) + 5 = {a^2}{b^3} + 3{a^2} + 5{b^3} + 15 + 5$

On adding the like terms, we get,

$\left( {{a^2} + 5} \right)\left( {{b^3} + 3} \right) + 5 = {a^2}{b^3} + 3{a^2} + 5{b^3} + 20$

iii. $\left( {t + {s^2}} \right)\left( {{t^2} - s} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

\[\left( {t + {s^2}} \right)\left( {{t^2} - s} \right) = t\left( {{t^2} - s} \right) + {s^2}\left( {{t^2} - s} \right)\]

\[\left( {t + {s^2}} \right)\left( {{t^2} - s} \right) = {t^3} - st + {s^2}{t^2} - {s^3}\]

iv. $\left( {a + b} \right)\left( {c - d} \right) + \left( {a - b} \right)\left( {c + d} \right) + 2\left( {ac + bd} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {a + b} \right)\left( {c - d} \right) + \left( {a - b} \right)\left( {c + d} \right) + 2\left( {ac + bd} \right) = a\left( {c - d} \right) + b\left( {c - d} \right) + a\left( {c + d} \right) - b\left( {c + d} \right) + 2ac + 2bd$

$\left( {a + b} \right)\left( {c - d} \right) + \left( {a - b} \right)\left( {c + d} \right) + 2\left( {ac + bd} \right) = ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd$

On grouping together and adding the like terms, we get,

$\left( {a + b} \right)\left( {c - d} \right) + \left( {a - b} \right)\left( {c + d} \right) + 2\left( {ac + bd} \right) = \left( {ac + ac + 2ac} \right) + \left( {ad - ad} \right) + \left( {bc - bc} \right) + \left( {2bd - bd - bd} \right)$

$\left( {a + b} \right)\left( {c - d} \right) + \left( {a - b} \right)\left( {c + d} \right) + 2\left( {ac + bd} \right) = 4ac$

v. $\left( {x + y} \right)\left( {2x + y} \right) + \left( {x + 2y} \right)\left( {x - y} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {x + y} \right)\left( {2x + y} \right) + \left( {x + 2y} \right)\left( {x - y} \right) = x\left( {2x + y} \right) + y\left( {2x + y} \right) + x\left( {x - y} \right) + 2y\left( {x - y} \right)$

$\left( {x + y} \right)\left( {2x + y} \right) + \left( {x + 2y} \right)\left( {x - y} \right) = 2{x^2} + xy + 2xy + {y^2} + {x^2} - xy + 2yx - 2{y^2}$

On grouping together and adding the like terms, we get,

$\left( {x + y} \right)\left( {2x + y} \right) + \left( {x + 2y} \right)\left( {x - y} \right) = \left( {2{x^2} + {x^2}} \right) + \left( {{y^2} - 2{y^2}} \right) + \left( {xy + 2xy - xy + 2xy} \right)$

$\left( {x + y} \right)\left( {2x + y} \right) + \left( {x + 2y} \right)\left( {x - y} \right) = 3{x^2} - {y^2} + 4xy$

vi. $\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right) = x\left( {{x^2} - xy + {y^2}} \right) + y\left( {{x^2} - xy + {y^2}} \right)$

$\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right) = {x^3} - {x^2}y + {y^2}x + y{x^2} - x{y^2} + {y^3}$

On grouping together and adding the like terms, we get,

$\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right) = {x^3} + {y^3} + \left( {x{y^2} - x{y^2}} \right) + \left( {{x^2}y - {x^2}y} \right)$

$\left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right) = {x^3} + {y^3}$

vii. $\left( {1.5x - 4y} \right)\left( {1.5x + 4y + 3} \right) - 4.5x + 12y$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {1.5x - 4y} \right)\left( {1.5x + 4y + 3} \right) - 4.5x + 12y = 1.5x\left( {1.5x + 4y + 3} \right) - 4y\left( {1.5x + 4y + 3} \right) - 4.5x + 12y$

$\left( {1.5x - 4y} \right)\left( {1.5x + 4y + 3} \right) - 4.5x + 12y = 2.25{x^2} + 6xy + 4.5x - 6xy - 16{y^2} - 12y - 4.5x + 12y$

On grouping together and adding the like terms, we get,

\[\left( {1.5x - 4y} \right)\left( {1.5x + 4y + 3} \right) - 4.5x + 12y = 2.25{x^2} + \left( {6xy - 6xy} \right) + \left( {4.5x - 4.5x} \right) - 16{y^2} + \left( {12y - 12y} \right)\]

\[\left( {1.5x - 4y} \right)\left( {1.5x + 4y + 3} \right) - 4.5x + 12y = 2.25{x^2} - 16{y^2}\]

viii. $\left( {a + b + c} \right)\left( {a + b - c} \right)$

Ans: We will multiply the given terms by applying the distributive property and then simplify the equation further. 

$\left( {a + b + c} \right)\left( {a + b - c} \right) = a\left( {a + b - c} \right) + b\left( {a + b - c} \right) + c\left( {a + b - c} \right)$

$\left( {a + b + c} \right)\left( {a + b - c} \right) = {a^2} + ab - ac + ab + {b^2} - bc + ac + bc - {c^2}$

On grouping together and adding the like terms, we get,

$\left( {a + b + c} \right)\left( {a + b - c} \right) = {a^2} + {b^2} - {c^2} + \left( {ab + ab} \right) + \left( {bc - bc} \right) + \left( {ac - ac} \right)$

$\left( {a + b + c} \right)\left( {a + b - c} \right) = {a^2} + {b^2} - {c^2} + 2ab$

NCERT Solutions for Class 8 Maths Chapter 8 Algebraic Expressions and Identities Exercise 8.4

Opting for the NCERT solutions for Ex 8.4 Class 8 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 8.4 Class 8 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

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