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# Polynominals: watch

1. I'm having a bit of a problem with this question:

The polynominal p(x), defined by:

p(x) = (x+2)^4 - 13x +a

has (x-1) as a factor.

(i) Find the value of the constant a

(ii) Verify that the remainder when p(x) is divided by (x+1) is (-54)

(iii) Find the quotient when p(x) is divided by (x-1)(x+4)

How do you even start the questions???
2. factor (or remainder) theorem:

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

i) P(1) = 0 => 3^4 - 13 + a = 0 => a = -68

ii) P(-1) = 1^4 +13 -68 = -54

iii) Q(x) = P(x)/(x-1) = [(x+2)^4 -13x -68]/(x-1)

= (x^4 + 8x³ + 24x² + 19x - 52)/(x-1) = x³ + 9x² + 33x + 52

remainder of Q(x)/(x+4) = Q(-4) = (-4)³ + 9(-4)² +33(-4) + 52 = 0

=> Q(x) can be divided by (x+4)

Q(x)/(x+4) = x² + 5x + 13
3. (Original post by Hexa)
I'm having a bit of a problem with this question:

The polynominal p(x), defined by:

p(x) = (x+2)^4 - 13x +a

has (x-1) as a factor.

(i) Find the value of the constant a

(ii) Verify that the remainder when p(x) is divided by (x+1) is (-54)

(iii) Find the quotient when p(x) is divided by (x-1)(x+4)

How do you even start the questions???
All questions of this sort can be solved by using the following:

if p(x) is a polynomial, it can also be expressed as q(x)(x-a) + R where q(x) is a polynomial with degree n-1 (if n is the degree of p(x)) and R is the remainder. Therefore, p(x) = R if x=a. So if (x-b) is a factor of any polynomial, then p(b)=0.

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