Question

use identity ( x + a ) ( x + b) = x² + ( a + b) x + ab to find the product of the following . (1) ( x + 3 ) ( x + 7 ) . (2) ( 2a² + a) ( 2a² + 5 ) ​

Answer 1

Question :

Find the product of the following . [ by using identity (x+a)(x+b)=x² + ( a + b) x + ab ]

1. ( x + 3 ) ( x + 7 )
2. ( 2a² + a) ( 2a² + 5 )

Solution :

We have to find the product

1) ( x + 3 ) ( x + 7 )

By using identity [ (x+a) (x+b)=x² + ( a + b) x + ab ]

$\sf\implies\:(x+3)(x+7)$

$\sf=x^2+(7+3)x+7\times3$

$\sf=x^2+10x+21$

Thus ,

( x + 3 ) ( x + 7 ) = x ²+ 10 x +21

2) ( 2a² + a) ( 2a² + 5 )

Now , use identity [ (x+a) (x+b)=x² + ( a + b) x + ab ]

Then ,

$\sf\implies(2a^2+a)(2a^2+5)$

$\sf=(2a^2)^2+(a+5)(2a^2)+5\times\:a$

$\sf=4a^4+2a^3+10a^2+5a$

_______________

More Algeraic Indentities:

1)$\sf{(a+b)^2=a^2+b^2+2ab}$

2)$\sf{(a-b)^2=a^2+b^2-2ab}$

3)$\sf{(a^2-b^2)=(a+b)(a-b)}$

$4)\sf{(a+b+c)^2={a}^{2}+{b}^{2}+{c}^{2}+2ab+2bc+2ca}$

5)$\sf{(a+b)^3=a^3+b^3+3ab(a+b)}$

Answer 2

${\bold{\sf{\underline{Understanding \: the \: question}}}}$

This question says that we have to use an identity ( x + a ) ( x + b) = x² + ( a + b) x + ab and now we have to find the product of ( x + 3 ) ( x + 7 ) and ( 2a² + a) ( 2a² + 5 )

${\bold{\sf{\underline{Full \: solution}}}}$

${\small{\red{\bold{\sf{Answer \: 1}}}}}$

Using property or identity

➥ ( x + a ) ( x + b) = x² + ( a + b) x + ab

➥ ( x + 3 ) ( x + 7 )

➥ x² + ( 7 + 3 )x + 7 × 3

➥ x² + 10x + 21

${\small{\red{\bold{\sf{Answer \: 2}}}}}$

Using property or identity

➥ ( x + a ) ( x + b) = x² + ( a + b) x + ab

➥ ( 2a² + a) ( 2a² + 5 )

➥ ( 2a² )² + ( a + 5 ) ( 2a² ) + 5 × a

➥ 4a⁴ + 2a³ + 10a² + 5a

${\bold{\sf{\underline{Knowledge \: booster}}}}$

Some algebraic identities -

$\boxed{\begin{minipage}{7 cm}\boxed{\bigstar\:\:\textbf{\textsf{Algebric\:Identity}}\:\bigstar}\\\\1)\bf\:(A+B)^{2} = A^{2} + 2AB + B^{2}\\\\2)\sf\: (A-B)^{2} = A^{2} - 2AB + B^{2}\\\\3)\bf\: A^{2} - B^{2} = (A+B)(A-B)\\\\4)\sf\: (A+B)^{2} = (A-B)^{2} + 4AB\\\\5)\bf\: (A-B)^{2} = (A+B)^{2} - 4AB\\\\6)\sf\: (A+B)^{3} = A^{3} + 3AB(A+B) + B^{3}\\\\7)\bf\:(A-B)^{3} = A^{3} - 3AB(A-B) + B^{3}\\\\8)\sf\: A^{3} + B^{3} = (A+B)(A^{2} - AB + B^{2})\\\\\end{minipage}}$

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