Polynomials

Question 1

Of what degree are the following?

i) P(x) = ax + b
ii) P(x) = 3x⁷ + 4x⁵ - 2

iii) P(x) = 6x^3/2 - 2/x + 4

Answer 1

i) Degree 1, linear
ii) Degree 7
iii) not a polynomial

Question 2

How would you write the following when performing long division?

P(x) = 3x⁷ + 4x⁵​ - 2

Answer 2

Any missing coefficients for powers of x are set to 0.

P(x) = 3x⁷ + 0x⁶ + 4x⁵ + 0x⁴ + 0x³ + 0x² - 2

Question 3

Describe the polynomial division algorithm.

Answer 3

Divide the first term in the polynomial by the first term in the divisor. This gives us the first term in the quotient (q₁)

Now multply q₁ by the divisor. Subtract this from the polynomial.

This should eliminate the first term in th polynomial.

Now bring down the next term.

Repeat.

P / D = Q + R

Polynomial / Divisor = Quotient + Remainder

Question 4

When can the remainder theorem be used?

Answer 4

When the denominator is linear.

e.g. (ax² + bx + c ) / (x+m)

Question 5

Describe the Reaminder theorem.

Answer 5

The linear remainder of P(x) when divided by (x - a).

R = P(a)

Useful for determining the remainder.

e.g.

P(x) = (x² + 9x + 4) / (x+1)

Dividing by x+1, so a = -1.

P(-1) = -1² + 9(-1) + 4

P(-1) = -4

This is the remainder

Question 6

Describe the factor theorem.

Answer 6

For P(x), if P(a) = 0, then (x - a) is a factor of P(x).

Therefore a is a zero or root of P(x).

if P(a) = 0, (x-a) is a factor.