Worksheet 4.7 Polynomials

Exercises:
So 3x 3 − 2x 2 + 4x + 7 divided by x 2 + 2x gives us a quotient of 3x − 8 with a
remainder of 20x + 7. We have
3x 3 − 2x 2 + 4x + 7 = (x 2 + 2x)(3x − 8) + (20x + 7).
More formally, suppose p(x) and f(x) are polynomials where deg p(x) ≥ deg f(x).
Then dividing p(x) by f(x) gives us the identity
p(x) = f(x)q(x) + r(x),
where q(x) is the quotient, r(x) is the remainder and deg r(x) < deg f(x).
Example 5 : Dividing p(x) = x 3 − 7x 2 + 4 by f(x) = x − 1 we obtain the
following result:
x 2 − 6x − 6
x − 1 x 3 − 7x 2 + 0x + 4
x 3 − x 2
−6x 2 + 0x
−6x 2 + 6x
−6x + 4
−6x + 6
−2
Here the quotient is q(x) = x 2 − 6x − 6 and the remainder is r = −2. Note: As
we can see, division doesn’t always produce a polynomial annswer- sometimes
there’s just a constant remainder.
1. Perform the following operations and find the degree of the result.
(a) (2x − 4x 2 + 7) + (3x 2 − 12x − 7)
(b) (x 2 + 3x)(4x 3 − 3x − 1)
(c) (x 2 + 2x + 1) 2
(d) (5x 4 − 7x 3 + 2x + 1) − (6x 4 + 8x 3 − 2x − 3)
2. Let p(x) = 3x 4 + 7x 2 − 10x + 4. Find p(1), p(0) and p(−2).
3. Carry out of the following divisions and write your answer in the form p(x) = f(x)q(x) + r(x).
(a) (3x 3 − x 2 + 4x + 7) ÷ (x + 2)
(b) (3x 3 − x 2 + 4x + 7) ÷ (x 2 + 2)
3