IB Mathematics Analysis and Approaches Topic AHL…

A polynomial $P(x)$ leaves remainders $3$ , $5$ and $11$ when divided by $x-1$ , and respectively.

(a) why the remainder when $P(x)$ is divided by $(x-1)(x-2)(x-3)$ has degree at most $2$ .

(b) Find that remainder.

Let $f(x)=3x^4+ax^3+bx^2+cx+d$ , where $a,b,c,d\in\mathbb{Z}$ , and suppose $x^2-x-6$ is a factor of $f(x)$ . Additionally, it is given that $f(2)=-72$ and the sum of the roots of $f(x)=0$ is $-2$ .

(a) Find the values of $a$ , $b$ , $c$ , and $d$ .

(b) Factorize $f(x)$ completely into linear factors over the real numbers.

Consider the polynomial $P(t)=t^4-5t^3+at^2+bt+c$ , where $a,b,c\in\mathbb{R}$ .

(a) If $(t+1)$ is a factor of $P(t)$ , an equation involving $a$ , $b$ , and $c$ .

(b) When $(t+1)^2$ is a factor of $P(t)$ , the value of $P'(-1)$ .

IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

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IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

IB Mathematics Analysis and Approaches AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 2.12—factor and Remainder Theorems, Sum and Product of Roots Questionbank

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