Think about a plan

Question

your friend multiplies x+4 by a quadratic polynomial and gets the result x^3-x^2-24x+30. The teacher says that everything is correct except of the constant term. Find the quadratic polynomial that your friend used. What is the correct result of multiplication?

• What does the face that all the terms except for the constant are correct tell you?
• How can polynomial division help you solve this problem?
• What is the connection between the remainder of the division and your friend's error?

Answer 1

A. The fact that everything is right except the constant means that we can get the correct polynomial by shifting the polynomial we have up or down.  Since (x+4) is supposed to be a factor, then we know that -4 should be a zero, which graphically shows us that there is one possible polynomial that meets the criteria.  We just need to figure out the right amount to change the constant by in order for -4 to be a zero of the polynomial.

 

B. If you do polynomial division, you'll end up with a quadratic and a constant remainder.  Using synthetic division, we get:

 

-4 | 1   -1   -24   30

    |     -4   -20    16

      1   -5   -4     46

 

The quadratic, x²-5x-4, can be multiplied by (x+4) in order to get a polynomial that is identical to the first except for the constant: x³-x²-24x-16.  The only difference between this and the original polynomial is that the constant is reduced by 46.  Therefore, x²-5x-4 is the quadratic that your friend used.

 

C. When we divide a polynomial by linear polynomial (x-a), the remainder is equal to the polynomial evaluated at a.  In this case, a = -4.  So our original polynomial is equal to 46 at x = -4.  You're welcome to verify this by substituting x = -4 into the original polynomial.  If we subtract 46 from this polynomial, then the new polynomial is now equal to 0 at x = -4, which is what we wanted.

 

Let me know if I lost you anywhere in that.