Question

Simplify:
\[\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25\]

Answer 1

Hint:
Here, we will multiply the polynomials by using the horizontal method or FOIL method to find the product. Then we will add the constant term to get the required answer. A polynomial expression is defined as an algebraic expression having one or more terms and these terms must be unlike.

Complete step by step solution:
The given algebraic expression is \[\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25\].
We will find the product of the given polynomials by FOIL method.
FOIL method is a method of multiplying the polynomials by multiplying the first terms, then the outer terms, then the inner terms and at last the last terms.
According to this method we will multiply one term of a polynomial with each term in the other polynomial.
Applying FOIL method on the expression, we get
\[\left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = \left( {{x^2} - 5} \right) \times x + \left( {{x^2} - 5} \right) \times 5 + 25\]
Now, using the distributive property of multiplication, we get
\[ \Rightarrow \left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = {x^2} \times x - 5x + 5{x^2} - 5 \times 5 + 25\]
On further multiplying the terms, we get
\[ \Rightarrow \left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = {x^3} - 5x + 5{x^2} - 25 + 25\]
Now, we will add and subtract the like terms in the above equation. Therefore, we get
\[ \Rightarrow \left( {{x^2} - 5} \right)\left( {x + 5} \right) + 25 = {x^3} + 5{x^2} - 5x\]
We can see that there are no more like terms, so this expression cannot be simplified further.

Therefore, the required simplified form of the given algebraic expression is equal to \[{x^3} + 5{x^2} - 5x\].

Note:
Here the obtained expression is a cubic expression. A cubic expression is an expression, where the highest degree of variable is 3. If an expression has a highest degree of variable as 2, we call it a quadratic expression. Similarly, if an expression has a highest degree of 1, then it is called a linear expression. Here we have also used the distributive property of multiplication. According distributive property of multiplication, if \[a\], \[b\] and \[c\] are three real numbers then \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].