Answer
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Hint:
The given polynomials are of degree four. We can factorise them using the identities in Algebra. Repeated applications may be needed for simplification.
Useful formula:
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
${(a + b)^2} = {a^2} + 2ab + {b^2}$
${(a - b)^2} = {a^2} - 2ab + {b^2}$
Complete step by step solution:
We have to factorise the given equations.
(i) ${a^4} - {b^4}$
We can write it as ${({a^2})^2} - {({b^2})^2}$.
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
So we have, ${({a^2})^2} - {({b^2})^2} = ({a^2} - {b^2})({a^2} + {b^2})$
We can again factorise using the same result.
This gives,
${({a^2})^2} - {({b^2})^2} = (a - b)(a + b)({a^2} + {b^2})$
So we have,
${a^4} - {b^4} = (a - b)(a + b)({a^2} + {b^2})$
(ii) ${p^4} - 81$
This can be written in the form ${({p^2})^2} - {9^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
This gives,
${({p^2})^2} - {9^2} = ({p^2} - 9)({p^2} + 9)$
We can again factorise using the same result for ${p^2} - 9$.
We have ${p^2} - 9 = {p^2} - {3^2}$
This gives,
${({p^2})^2} - {9^2} = (p - 3)(p + 3)({p^2} + 9)$
So we have,
${p^4} - 81 = (p - 3)(p + 3)({p^2} + 9)$
(iii) ${x^4} - {(y + z)^4}$
This expression can be rewritten as ${({x^2})^2} - {({(y + z)^2})^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
This gives,
${({x^2})^2} - {({(y + z)^2})^2} = ({x^2} - {(y + z)^2})({x^2} + {(y + z)^2})$
Again we can use the same formula for ${x^2} - {(y + z)^2}$.
We have, ${x^2} - {(y + z)^2} = (x - (y + z))(x + (y + z))$
$ \Rightarrow {x^2} - {(y + z)^2} = (x - y - z)(x + y + z)$
This gives,
${({x^2})^2} - {({(y + z)^2})^2} = (x - y - z)(x + y + z)({x^2} + {(y + z)^2})$
We have, ${(a + b)^2} = {a^2} + 2ab + {b^2}$.
We can use this result for ${(y + z)^2}$.
${({x^2})^2} - {({(y + z)^2})^2} = (x - y - z)(x + y + z)({x^2} + {y^2} + 2yz + {z^2})$
So we have,
${x^4} - {(y + z)^4} = (x - y - z)(x + y + z)({x^2} + {y^2} + 2yz + {z^2})$
(iv) ${x^4} - {(x - z)^4}$
We can write this as ${({x^2})^2} - {({(x - z)^2})^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
This gives,
${({x^2})^2} - {({(x - z)^2})^2} = ({x^2} - {(x - z)^2})({x^2} + {(x - z)^2})$
Again we can use the same formula for ${x^2} - {(x - z)^2}$.
${x^2} - {(x - z)^2} = (x - (x - z))(x + (x - z))$
Simplifying we get,
${x^2} - {(x - z)^2} = (x - x + z)(x + x - z)$
$ \Rightarrow {x^2} - {(x - z)^2} = z(2x - z)$
This gives,
${({x^2})^2} - {({(x - z)^2})^2} = z(2x - z)({x^2} + {(x - z)^2})$
We have, ${(a - b)^2} = {a^2} - 2ab + {b^2}$.
We can use this result for ${(x - z)^2}$.
$ \Rightarrow {({x^2})^2} - {({(x - z)^2})^2} = z(2x - z)({x^2} + ({x^2} - 2xz + {z^2}))$
Simplifying we get,
$ \Rightarrow {({x^2})^2} - {({(x - z)^2})^2} = z(2x - z)(2{x^2} - 2xz + {z^2})$
So we have,
${x^4} - {(x - z)^4} = z(2x - z)(2{x^2} - 2xz + {z^2})$
(v) ${a^4} - 2{a^2}{b^2} + {b^4}$
We can write it as ${({a^2})^2} - 2{a^2}{b^2} + {({b^2})^2}$
${a^4} - 2{a^2}{b^2} + {b^4} = {({a^2})^2} - 2{a^2}{b^2} + {({b^2})^2}$
We have, ${(a + b)^2} = {a^2} + 2ab + {b^2}$.
Comparing these two we have,
${a^4} - 2{a^2}{b^2} + {b^4} = {({a^2} - {b^2})^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
So we have,
${a^4} - 2{a^2}{b^2} + {b^4} = {[(a - b)(a + b)]^2}$
Note:
When an equation of degree two or more is given, we have to factorise maximum to linear factors. The main identities used are the expansions of ${(a + b)^2},{(a - b)^2}$ and ${a^2} - {b^2}$. In certain questions repeated application of the same result may be needed.
The given polynomials are of degree four. We can factorise them using the identities in Algebra. Repeated applications may be needed for simplification.
Useful formula:
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
${(a + b)^2} = {a^2} + 2ab + {b^2}$
${(a - b)^2} = {a^2} - 2ab + {b^2}$
Complete step by step solution:
We have to factorise the given equations.
(i) ${a^4} - {b^4}$
We can write it as ${({a^2})^2} - {({b^2})^2}$.
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
So we have, ${({a^2})^2} - {({b^2})^2} = ({a^2} - {b^2})({a^2} + {b^2})$
We can again factorise using the same result.
This gives,
${({a^2})^2} - {({b^2})^2} = (a - b)(a + b)({a^2} + {b^2})$
So we have,
${a^4} - {b^4} = (a - b)(a + b)({a^2} + {b^2})$
(ii) ${p^4} - 81$
This can be written in the form ${({p^2})^2} - {9^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
This gives,
${({p^2})^2} - {9^2} = ({p^2} - 9)({p^2} + 9)$
We can again factorise using the same result for ${p^2} - 9$.
We have ${p^2} - 9 = {p^2} - {3^2}$
This gives,
${({p^2})^2} - {9^2} = (p - 3)(p + 3)({p^2} + 9)$
So we have,
${p^4} - 81 = (p - 3)(p + 3)({p^2} + 9)$
(iii) ${x^4} - {(y + z)^4}$
This expression can be rewritten as ${({x^2})^2} - {({(y + z)^2})^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
This gives,
${({x^2})^2} - {({(y + z)^2})^2} = ({x^2} - {(y + z)^2})({x^2} + {(y + z)^2})$
Again we can use the same formula for ${x^2} - {(y + z)^2}$.
We have, ${x^2} - {(y + z)^2} = (x - (y + z))(x + (y + z))$
$ \Rightarrow {x^2} - {(y + z)^2} = (x - y - z)(x + y + z)$
This gives,
${({x^2})^2} - {({(y + z)^2})^2} = (x - y - z)(x + y + z)({x^2} + {(y + z)^2})$
We have, ${(a + b)^2} = {a^2} + 2ab + {b^2}$.
We can use this result for ${(y + z)^2}$.
${({x^2})^2} - {({(y + z)^2})^2} = (x - y - z)(x + y + z)({x^2} + {y^2} + 2yz + {z^2})$
So we have,
${x^4} - {(y + z)^4} = (x - y - z)(x + y + z)({x^2} + {y^2} + 2yz + {z^2})$
(iv) ${x^4} - {(x - z)^4}$
We can write this as ${({x^2})^2} - {({(x - z)^2})^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
This gives,
${({x^2})^2} - {({(x - z)^2})^2} = ({x^2} - {(x - z)^2})({x^2} + {(x - z)^2})$
Again we can use the same formula for ${x^2} - {(x - z)^2}$.
${x^2} - {(x - z)^2} = (x - (x - z))(x + (x - z))$
Simplifying we get,
${x^2} - {(x - z)^2} = (x - x + z)(x + x - z)$
$ \Rightarrow {x^2} - {(x - z)^2} = z(2x - z)$
This gives,
${({x^2})^2} - {({(x - z)^2})^2} = z(2x - z)({x^2} + {(x - z)^2})$
We have, ${(a - b)^2} = {a^2} - 2ab + {b^2}$.
We can use this result for ${(x - z)^2}$.
$ \Rightarrow {({x^2})^2} - {({(x - z)^2})^2} = z(2x - z)({x^2} + ({x^2} - 2xz + {z^2}))$
Simplifying we get,
$ \Rightarrow {({x^2})^2} - {({(x - z)^2})^2} = z(2x - z)(2{x^2} - 2xz + {z^2})$
So we have,
${x^4} - {(x - z)^4} = z(2x - z)(2{x^2} - 2xz + {z^2})$
(v) ${a^4} - 2{a^2}{b^2} + {b^4}$
We can write it as ${({a^2})^2} - 2{a^2}{b^2} + {({b^2})^2}$
${a^4} - 2{a^2}{b^2} + {b^4} = {({a^2})^2} - 2{a^2}{b^2} + {({b^2})^2}$
We have, ${(a + b)^2} = {a^2} + 2ab + {b^2}$.
Comparing these two we have,
${a^4} - 2{a^2}{b^2} + {b^4} = {({a^2} - {b^2})^2}$
For any $a,b$ we have, ${a^2} - {b^2} = (a - b)(a + b)$
So we have,
${a^4} - 2{a^2}{b^2} + {b^4} = {[(a - b)(a + b)]^2}$
Note:
When an equation of degree two or more is given, we have to factorise maximum to linear factors. The main identities used are the expansions of ${(a + b)^2},{(a - b)^2}$ and ${a^2} - {b^2}$. In certain questions repeated application of the same result may be needed.
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