Dividing Polynomials; Remainder and Factor Theorems

Question 1

What’s the division algorithm when dividing a polynomial by a polynomial.

Answer 1

1. Arrange both the dividend and divisor terms in descending order of variable powers.
2. Divide the dividend’s first term by the divisor’s first term to get the first term of the quotient.
3. Multiply the entire divisor by the quotient’s first term. Write this product beneath the dividend, aligning like terms.
4. Subtract this product from the dividend.
5. Write the next term from the original dividend next to the remainder to create a new dividend.
6. Repeat this process with the new dividend until you can’t divide further—when the remainder’s degree (highest variable exponent) becomes less than the divisor’s degree. **This will occur when the degree of the remainder(the highest exponent on a variable in the remainder) is less than the degree of the divisor. If the final dividend is not 0, write the dividend over the divisor in the quotient.

Question 2

What’s synthetic division?

Answer 2

Synthetic division is a shorthand way of doing polynomial division in the special case of dividing by a linear factor. The linear factor must always be in the form x - c.

Question 3

Describe how synthetic division works.

Answer 3

Here’s the algorithm for how this method works.
1. Setup the synthetic division block.
2. Bring the first coefficient down, and multiply it by the constant term of x - c. Write down the result below the second coefficient and sum the two numbers, write the sum under the line, and repeat until the last coefficient value.
3. Write the variables by starting with n - 1 – n is the degree of the polynomial – as your exponential value for the variable of the first coefficient, all the way down to the constant term of the new quotient, if there’s another number after the term with variable x^0 then this is the reminder and it should be written b/(divisor).

Question 4

Can this expression be divided using synthetic division: (x^2 + 3x + 1)/(2x - 3)?

Answer 4

No, because the divisor is not in the form x - c, even though it is linear.

Question 5

a - quotient
b - divisor
c - dividend
d - remainder

How is c found if a, b, and r, are availabe?

Answer 5

c = ab + r.

Question 6

What does the Remainder Theorem state?

Answer 6

The Remainder Theorem states that if a polynomial f(x) is divided by a linear divisor of the form x - c, then the remainder of that division is equal to f(c). In other words, the remainder is the value of the polynomial when evaluated at x = c.

Question 7

What does the Factor Theorem State?

Answer 7

It states that if f(x) is a polynomial divided by x - c, and f(c) = 0, then x - c is a factor of f(x).