Synthetic Substitution-Using the Remainder Theorem

Welcome to MooMooMath
Today we are going to look at synthetic substitution which is also known as the math remainder theorem
You can substitute values into a polynomial by taking the function of 2 in function notation which means for the function everywhere there is an x you will substitute a 2
Now we can also use synthetic substitution to evaluate the polynomial.
If we want to evaluate this equation for the number 2
We can plug in a 2 and evaluate or we can use synthetic division.
What we do is take the value 2 and basically divide.
We take the coefficient which 1 and we don't have a cubic term so we have a zero. The quadratic term is two and the linear term is 1 and the constant is 5
We bring down the 1 multiply by 2 and add the column now multiply by 2 and add the column then multiply by 2 which equals 12 multiply by 2 and add the column and you get 27
This tells us we have a polynomial with a remainder of 27.
What we want to look at is the remainder.
We don't use the polynomial.
It tells us when we plug 2 into the function our output is 27
We then write this in function notation.
The function evaluated at 2 is 27
This is how you synthetic substitute and the remainder theorem tells us when we use two we get an output of 27
Let work another problem.
Take two and plug it into the function.
2 to the fourth plus 2 squared minus 2 plus 5
2 to the fourth is 16 plus 4 times 2 is 8 minus 2 plus 5 equals 24 minus 2 equals 22 plus 5 equals 27
You can evaluate the function or use synthetic substitution
where you divide the polynomial by the value and your remainder is your output.