Question

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

Answer 1

Hint:We have multiplication of two quadratic polynomials in variables p and q that is $\left( {{p^2} - {q^2}} \right)$ and $\left( {{p^2} + {q^2}} \right)$. Both the polynomials have degree one and have two terms each. Hence, both polynomials are called binomials and linear polynomials. To multiply polynomials, first multiply each term in one polynomial by each term in the other polynomial using distributive law. Then, simplify the resulting polynomial by adding or subtracting the like terms.

Complete step by step answer:
So, we have, $\left( {{p^2} - {q^2}} \right)\left( {{p^2} + {q^2}} \right)$. In the first polynomial, we have two square terms and in the second polynomial also we have two terms of degree two. Multiply the first term of a polynomial with second polynomial and then the second term with second polynomial, we have,
$ \Rightarrow {p^2}\left( {{p^2} + {q^2}} \right) - {q^2}\left( {{p^2} + {q^2}} \right)$
Opening the brackets and multiplying, we have,
$ \Rightarrow {p^4} + {p^2}{q^2} - {q^2}{p^2} - {q^4}$
Cancelling the like terms with opposite signs, we get,
$ \Rightarrow {p^4} - {q^4}$

Thus we have, $\left( {{p^2} - {q^2}} \right)\left( {{p^2} + {q^2}} \right) = {p^4} - {q^4}$.

Additional information: Degree of a polynomial is the highest of the degrees of the individual term with non-zero coefficients. We have different types polynomials based on their degree such as: constant polynomial, linear polynomial, quadratic polynomial, cubic polynomial and quartic polynomial etc. in this case, we were given the product of two quadratic polynomial in variables p and q.

Note: For avoiding mistakes, write the terms in the decreasing order of their exponent. Thus we obtained a polynomial of degree two. Hence, the obtained polynomial is a quadratic polynomial. When we multiply two polynomials of any degree the obtained polynomial must have degree higher than multiplied individual polynomials. We can check this in the given above problem and it satisfies the condition. Be careful with the sign when you open the brackets. Follow the same procedure for any two polynomials.