Some examples on cubic identities — lesson. Mathematics CBSE, Class 9.

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Let us expand some of the cubic terms using their identities.

$(2x+3y)^3$

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.

$(2x+3y)^3$

$=$

$(2x)^3$

$+$

$3(2x)^2(3y)$

$+$

$3(2x)(3y)^2$

$+$

$(3y)^3$

$(2x+3y)^3$

$=$

$8x^3$

$+$

$(3 \times 4 \times 3)x^2y$

$+$

$(3\times 2\times 9)xy^2$

$+$

$27y^3$

$=$

$8x^3 + 36x^2y + 54xy^2 + 27y^3$

$(5x-7y)^3$

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

$a=5x$ and

$b=7y$ .

Substitute the values in the formula.

$(5x-7y)^3$

$=$

$(5x)^3$

$-$

$3(5x)^2(7y)$

$+$

$3(5x)(7y)^2$

$-$

$(7y)^3$

$(5x-7y)^3$

$=$

$125x^3$

$-$

$(3\times 25\times 7)x^2y$

$+$

$(3\times 5 \times 49)xy^2$

$-$

$343y^3$

$(5x-7y)^3$

$=$

$125x^3$

$-$

$525x^2y$

$+$

$735xy^2$

$-$

$343y^3$

Example:

Look for the following cases where we used the identities.

1. Expand

$(y-5)^3$ using the identity.

The above expression is of the form

$(a-b)^3$ .

We have the identity,

$(a-b)^3$

$=$

$a^3-3a^2b+3ab^2-b^3$ .

Substitute

$a = y$ and

$b = 5$ in the formula.

${(y - 5)}^{3} = y^{3} - 3 {(y)}^{2} (5) + 3 (y) {(5)}^{2} - 5^{3}$

${(y - 5)}^{3} = y^{3} - 15 y^{2} + 75 y - 125$

2. Evaluate

$103^3$ using the identity.

Rewrite

$103^3$ as

$(100+3)^3$ .

The above expression is of the form

$(a+b)^3$ .

We have the identity,

$(a+b)^3$

$=$

$a^3+3a^2b+3ab^2+b^3$

Substitute

$a =100$ and

$b = 3$ in the formula.

${(100 + 3)}^{2} = 100^{3} + 3 {(100)}^{2} (3) + 3 (100) {(3)}^{2} + 3^{3}$

$= 1000000 + (3 \times 10000 \times 3) + (3 \times 100 \times 9) + 27$

$= 1000000 + 90000 + 2700 + 27$

$= 1092727$

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3. Some examples on cubic identities

Theory: