Synthetic Division Help
Introduction to Synthetic Division
Synthetic division of polynomials is much easier than long division. It only works when the divisor is of a certain form, though. Here, we will use synthetic division when the divisor is of the form “ x − number” or “ x + number.”
For a problem of the form
Every power of x must be represented.
In synthetic division, the tedious work in long division is reduced to a few steps.
Find the quotient and remainder using synthetic division.
Bring down the first coefficient.
Multiply this coefficient by 2 (the c ) and put the product under −5, the next coefficient.
Add −5 and 8. Put the sum under 8.
Multiply 3 by 2 and put the product under 1, the next coefficient.
Add 1 and 6. Put the sum under 6.
Multiply 7 by 2. Put the product under −8, the last coefficient.
Add −8 and 14. Put the sum under 14. This is the last step.
The numbers on the last row are the coefficients of the quotient and the remainder. The remainder is a constant (which is a term of degree 0), and the degree of the quotient is exactly one less degree than the degree of the dividend. In this example, the degree of the dividend is 3, so the degree of the quotient is 2. The last number on the bottom row is the remainder. The numbers before it are the coefficients of the quotient, in order from the highest degree to the lowest. The remainder in this example is 6. The coefficients of the quotient are 4, 3, and 7. The quotient is 4 x 2 + 3 x + 7.
(3 x 4 − x 2 + 2 x + 9) ÷ ( x + 5)
Because x + 5 = x − (−5), c = −5.
Bring down 3, the first coefficient. Multiply it by −5. Put 3(−5) = −15 under 0.
Add 0 + (−15) = −15. Multiply −15 by −5 and put (−15) (−5) = 75 under − 1.
Add −1 and 75. Multiply −1 + 75 = 74 by −5 and put (74)(−5) = −370 under 2.
Add 2 to −370. Multiply 2 + (−370) = −368 by −5 and put (−368) (−5) = 1840 under 9.
Add 9 to 1840. Put 9 + 1840 = 1849 under 1840.
The dividend has degree 4, so the quotient has degree 3. The quotient is 3 x 3 − 15 x 2 + 74 x − 368 and the remainder is 1849.
The Remainder Theorem
When dividing a polynomial f(x) by x − c , the remainder tells us two things. If we get a remainder of 0, then both the divisor, ( x − c ), and quotient are factors of f(x) . Another fact we get from the remainder is that f(c) = remainder.
f(x) = (x − c)q(x) + remainder
f(c) = (c − c)q(c) + remainder Evaluate f(x) at x = c .
f(c) = 0q(c) + remainder
f(c) = remainder
The fact that f(c) is the remainder is called the Remainder Theorem . It is useful when trying to evaluate complicated polynomials. We can also use this fact to check our work with synthetic division and long division (providing the divisor is x − c ).
( x 3 - 6 x 2 + 4 x −5) ÷ ( x − 3)
By the Remainder Theorem, we should get the remainder to be 3 3 − 6(3 2 ) + 4(3) − 5 = −20.
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