UPSKILL MATH PLUS
Learn Mathematics through our AI based learning portal with the support of our Academic Experts!
Learn moreLet us expand some of the cubic terms using its identities.
Write the cube in expanded form.
1. (2x+3y)^3
Let us use the identity, (a+b)^3= a^3+3a^2b+3ab^2+b^3.
Comparing (2x+3y)^3 with (a+b)^3, we have a=2x and b=3y.
Substitute the values in the formula.
(2x+3y)^3 = (2x)^3+3(2x)^2(3y)+3(2x)(3y)^2+(3y)^3
(2x+3y)^3 = 8x^3+(2\times 4\times 3)x^2y+(3\times 2\times 9)xy^2+27y^3
= 8x^3+24x^2y+54xy^2+27y^3
2. (5x-7y)^3
Let us use the identity, (a-b)^3=a^3-3a^2b+3ab^2-b^3.
Comparing (5x-7y)^3 with (a-b)^3, we have a=5x an db=7y.
Substitute the values in the formula.
(5x-7y)^3 = (5x)^3-3(5x)^2(7y)+3(5x)(7y)^2+(7y)^3
(5x-7y)^3 = 125x^3-(3\times 25\times 7)x^2y+(3\times 5 \times 49)xy^2+343y^3
(5x-7y)^3 = 125x^3-525x^2y+735xy^2+343y^3
4. (4y+5)(4y+3)(4y-7)
Let us use the identity, (x+a)(x+b)(x+c) = x^3+(a+b+c)x^2+(ab+bc+ca)x+abc
Comparing (4y+5)(4y+3)(4y-7) with (x+a)(x+b)(x+c), we have x=4y, a=5, b=3 an dc=-7.
Substitute the known values.
(4y+5)(4y+3)(4y-7) = (4y)^3+(5+3-7)(4y)^2+((5\times 3) + (3\times -7) +(-7\times 5))(4y)+5 \times 3 \times -7)
(4y+5)(4y+3)(4y-7) = 64y^3+16y^2+(15-21-35)(4y)-105
(4y+5)(4y+3)(4y-7) = 64y^3+16y^2-164y-105
Example:
Look for the following cases where we used the identities.
1. Expand (y-5)^3 using identity.
The above expression is in (a-b)^3 form.
We have the identity, (a-b)^3=a^3-3a^2b+3ab^2-b^3.
Substitute a = y and b = 5 in the formula.
2. Evaluate 103^3 using identity.
103^3 = (100+3)^3
The above expression is in (a+b)^3 form.
We have the identity, (a+b)^3 = a^3+3a^2b+3ab^2+b^3
Substitute a =100 and b = 3 in the formula.
Register for free to see more content