Part 2: Synthetic Division & The Remainder Theorem

Synthetic DivisionSynthetic Division is a process that

simplifies long division, but it can only be used when dividing a polynomial by a linear factor of the form x – a.

Synthetic Division1. Write the polynomial in standard form,

including zero coefficients where appropriate2. Set up: use the opposite sign of a (this allows

us to add throughout the process) and write the coefficients of the polynomial.

3. Bring down the first coefficient4. Multiply the coefficient by the divisor. Add to

the next coefficient.5. Continue multiplying and adding through the

last coefficient.6. Write the quotient and remainder. The

remainder will be the last sum.

Example: Divide using synthetic division

3 57 56 7x x x

Example: Divide using synthetic division

3 214 51 54 2x x x x

Example: Using synthetic division to solve a problemThe polynomialexpresses the volume, in cubic inches, of the shadow box shown.

3 27 38 240x x x

1.What are the dimensions of the box?Hint: the length is greater than the height (or depth)

2.If the width of the box is 15 in, what are the other dimension?

The Remainder TheoremThe Remainder Theorem provides a quick

way to find the remainder of a polynomial long-division problem.

If you divide a polynomial P(x) of degree by then the remainder is P(a)

1nx a

Example: Evaluating a PolynomialGiven that

what is P(3)?By the remainder theorem, P(3) is the

remainder when you divide P(x) by x – 3.

5 3 2( ) 2 2P x x x x

Example: Evaluating a PolynomialGiven that

what is P(─ 4)?By the remainder theorem, P(─ 4) is the

remainder when you divide P(x) by x + 4.

5 4 3( ) 3 28 5 20P x x x x x

HomeworkP308 #21 – 39 odd, 40 – 43 all, 53 – 56 all, 57

– 61 odd