Question

If j(x) = x4 - 81, and k(x) = x + 3 then j(x) k(x) =

Answer 1

Hi there!
The question here is asking us to multiply two functions together - j(x) and k(x). First, we need to determine the expressions for j(x) and k(x). Since this is given, we can move straight onto multiplying the two functions together, which will give us our answer.
j(x) × k(x)
Substitute expressions - 
(x⁴ - 81)(x + 3)
Simplify - 
(x⁴ - 81)(x + 3)
x⁵ + 3x⁴ - 81x - 243
Therefore, the answer is x⁵ + 3x⁴ - 81x - 243. Hope this helped!

Answer 2

Trying to factor as a Difference of Squares :

 1.1      Factoring:  x4-81 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2 - AB + AB - B2 = 
         A2 - B2

Note :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 81 is the square of 9
Check :  x4  is the square of  x2 

Factorization is :       (x2 + 9)  •  (x2 - 9) 

Polynomial Roots Calculator :

 1.2    Find roots (zeroes) of :       F(x) = x2 + 9
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q  then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  1  and the Trailing Constant is  9. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,9 

 Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      10.00        -3     1      -3.00      18.00        -9     1      -9.00      90.00        1     1      1.00      10.00        3     1      3.00      18.00        9     1      9.00      90.00   

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 1.3      Factoring:  x2 - 9 

Check : 9 is the square of 3
Check :  x2  is the square of  x1 

Factorization is :       (x + 3)  •  (x - 3) 

Final result : (x2 + 9) • (x + 3) • (x - 3)