LFM Stats And Pure

kyotsu-test 2012 QCourse2-I-Q2 Factorization and Root Analysis View

Consider the polynomial
$$P = x^2 + 2(a-1)x - 8a - 8.$$
(1) Let $a$ be a rational number. If the value of $P$ is a rational number when $x = 1 - \sqrt{2}$, then $a =$ $\mathbf{K}$ and in this case the value of $P$ is $P =$ $\mathbf{LM}$.
(2) Let $x$ and $a$ be positive integers. We are to investigate $x$ and $a$ which are such that the value of $P$ is a prime number.
When we factorize $P$, we have
$$P = (x - \mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$
Hence $x$ must be $\mathbf{Q}$.
Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.

kyotsu-test 2015 QCourse1-I-Q1 Factorization and Root Analysis View

Set $P = 10a^2 + 14ab - 21bc - 15ca$.
(1) Factorizing $P$, we obtain $$P = (\mathbf{A}a + \mathbf{B}b)(\mathbf{C}a - \mathbf{D}c).$$
(2) If $5a = \sqrt{6}$, $14b = \sqrt{2} + \sqrt{3} - \sqrt{6}$ and $15c = \sqrt{12} - \sqrt{8}$, then $$P = \frac{\mathbf{E}}{\mathbf{E} + \mathbf{F}} \frac{\mathbf{G}}{\mathbf{H}}$$ and hence the greatest integer less than $P$ is $\mathbf{I}$.

mat 2001 Q2 15 marks Factorization and Root Analysis View

(a) Factorize the expression $x ^ { 2 } + 3 x - 10$.
(b) If $x ^ { 3 } + a x ^ { 2 } + b x + c = ( x - \alpha ) ( x - \beta ) ( x - \gamma )$ for all values of $x$, find $a , b , c$ in terms of $\alpha , \beta , \gamma$.
(c) Find a value of $b$ for which $x ^ { 3 } + b x + 2 = 0$ has exactly two distinct solutions.

taiwan-gsat 2006 Q9 Sum of Coefficients and Coefficient Relationships View

9. A student practices calculating the remainder when a cubic polynomial $f ( x )$ is divided by a linear polynomial $g ( x )$. It is known that the coefficient of the cubic term of $f ( x )$ is 3, and the coefficient of the linear term is 2. Student A mistakenly read the coefficient of the cubic term of $f ( x )$ as 2 (other coefficients were read correctly), and Student B mistakenly read the coefficient of the linear term of $f ( x )$ as $- 2$ (other coefficients were read correctly). The remainders calculated by Student A and Student B happen to be the same. Which of the following linear expressions could $g ( x )$ equal?
(1) $x$
(2) $x - 1$
(3) $x - 2$
(4) $x + 1$
(5) $x + 2$

taiwan-gsat 2008 Q10 True/False or Multiple-Statement Evaluation View

10. Let $f ( x )$ and $g ( x ) = x ^ { 3 } + x ^ { 2 } - 2$ be real coefficient polynomials with a common factor of degree greater than 0. Which of the following statements are correct?
(1) $g ( x ) = 0$ has exactly one real root
(2) $f ( x ) = 0$ must have a real root
(3) If $f ( x ) = 0$ and $g ( x ) = 0$ have a common real root, then this root must be 1
(4) If $f ( x ) = 0$ and $g ( x ) = 0$ have a common real root, then the greatest common divisor of $f ( x )$ and $g ( x )$ is a linear polynomial
(5) If $f ( x ) = 0$ and $g ( x ) = 0$ have no common real roots, then the greatest common divisor of $f ( x )$ and $g ( x )$ is a quadratic polynomial

taiwan-gsat 2009 Q3 True/False or Multiple-Statement Evaluation View

3. Given that $f(x)$ and $g(x)$ are two real-coefficient polynomials, and the remainder when $f(x)$ is divided by $g(x)$ is $x^{4} - 1$. Which of the following options cannot be a common factor of $f(x)$ and $g(x)$?
(1) 5
(2) $x - 1$
(3) $x^{2} - 1$
(4) $x^{3} - 1$
(5) $x^{4} - 1$

taiwan-gsat 2021 Q5 8 marks True/False or Multiple-Statement Evaluation View

Assume $f ( x )$ is a fifth-degree polynomial with real coefficients, and the remainder when $f ( x )$ is divided by $x ^ { n } - 1$ is $r _ { n } ( x )$ , where $n$ is a positive integer. Select the correct options.
(1) $r _ { 1 } ( x ) = f ( 1 )$
(2) $r _ { 2 } ( x )$ is a first-degree polynomial with real coefficients
(3) The remainder when $r _ { 4 } ( x )$ is divided by $x ^ { 2 } - 1$ equals $r _ { 2 } ( x )$
(4) $r _ { 5 } ( x ) = r _ { 6 } ( x )$
(5) If $f ( - x ) = - f ( x )$ , then $r _ { 3 } ( - x ) = - r _ { 3 } ( x )$

taiwan-gsat 2021 Q5 5 marks Remainder by Quadratic or Higher Divisor View

Let $f ( x )$ be a real polynomial function of degree 3 satisfying the condition that the remainder when $( x + 1 ) f ( x )$ is divided by $x ^ { 3 } + 2$ is $x + 2$. If $f ( 0 ) = 4$, what is the value of $f ( 2 )$?
(1) 8
(2) 10
(3) 15
(4) 18
(5) 20

taiwan-gsat 2022 Q4 8 marks Divisibility and Factor Determination View

Let the polynomials $f(x) = x^3 + 2x^2 - 2x + k$ and $g(x) = x^2 + ax + 1$, where $k, a$ are real numbers. Given that $g(x)$ divides $f(x)$ and the equation $g(x) = 0$ has complex roots, select the option that is a root of the equation $f(x) = 0$.
(1) $-3$
(2) $0$
(3) $1$
(4) $\frac{1 + \sqrt{-3}}{2}$
(5) $\frac{3 + \sqrt{-5}}{2}$

taiwan-gsat 2023 Q6 5 marks Sum of Coefficients and Coefficient Relationships View

A person calculates the remainder when the polynomial $f(x) = x^{3} + ax^{2} + bx + c$ is divided by $g(x) = ax^{3} + bx^{2} + cx + d$, where $a, b, c, d$ are real numbers and $a \neq 0$. He mistakenly read it as $g(x)$ divided by $f(x)$, and after calculation obtained the remainder as $-3x - 17$. Assuming the correct remainder when $f(x)$ is divided by $g(x)$ equals $px^{2} + qx + r$, what is the value of $p$?
(1) $-3$ (2) $-1$ (3) $0$ (4) $2$ (5) $3$

taiwan-gsat 2023 Q14 5 marks Remainder by Quadratic or Higher Divisor View

Let $a$ and $b$ be real numbers (where $a > 0$). If the polynomial $ax^{2} + (2a+b)x - 12$ divided by $x^{2} + (2-a)x - 2a$ gives a remainder of 6, then the ordered pair $(a, b) = $ (14--1), 14--2).

taiwan-gsat 2024 Q5 8 marks True/False or Property Verification Statements View

Let $f(x)$ be a cubic polynomial with real coefficients. It is known that $f(-2 - 3i) = 0$ (where $i = \sqrt{-1}$), and the remainder when $f(x)$ is divided by $x^{2} + x - 2$ is 18. Select the correct options.
(1) $f(2 + 3i) = 0$
(2) $f(-2) = 18$
(3) The coefficient of the cubic term of $f(x)$ is negative
(4) $f(x) = 0$ has exactly one positive real root
(5) The center of symmetry of the graph $y = f(x)$ is in the first quadrant

taiwan-gsat 2024 Q9 5 marks Euclidean Division: Quotient and Remainder Determination View

Given that when polynomial $f ( x )$ is divided by $x ^ { 2 } + 5 x + 1$, the quotient is $x ^ { 3 } + 7 x ^ { 2 } + x + 3$, select the options that could be $f ( x )$.
(1) $2 \left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right)$
(2) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x$
(3) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right) + x ^ { 2 }$
(4) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 4 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x$
(5) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 4 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x ^ { 2 }$

taiwan-gsat 2024 Q14 5 marks Remainder by Quadratic or Higher Divisor View

It is known that $f(x), g(x), h(x)$ are all real-coefficient cubic polynomials, and their remainders when divided by $x^{2} - 2x + 3$ are $x + 1$, $x - 3$, and $-2$ respectively. If $xf(x) + ag(x) + bh(x)$ is divisible by $x^{2} - 2x + 3$, where $a, b$ are real numbers, then $a =$ (14－1)(14－2), $b =$ (14－3).

taiwan-gsat 2025 Q13 5 marks Euclidean Division: Quotient and Remainder Determination View

A real-coefficient cubic polynomial $f ( x )$ divided by $x + 6$ gives quotient $q ( x )$ and remainder 3. If $q ( x )$ has a maximum value of 8 at $x = - 6$, then the coordinates of the center of symmetry of the graph $y = f ( x )$ are ((13-1) (13-2), (13-3)).

tmua None Q6 Divisibility and Factor Determination View

6. It is given that $x + 2$ is a factor of $x ^ { 3 } + 4 c x ^ { 2 } + x ( c + 1 ) ^ { 2 } - 6$.
The sum of the possible values of $c$ is
A - 10
B - 6
C 0
D 6
E 10

tmua None Q12 Remainder by Linear Divisor View

12. A polynomial $p ( x )$ has the property that $p ( 1 ) = 2$.
Which one of the following can be deduced from this?
A $\quad p ( x ) = ( x - 1 ) q ( x ) + 2$ for some polynomial $q ( x )$.
B $\quad p ( x ) = ( x + 1 ) q ( x ) + 2$ for some polynomial $q ( x )$.
C $\quad p ( x ) = ( x - 1 ) q ( x ) - 2$ for some polynomial $q ( x )$.
D $\quad p ( x ) = ( x + 1 ) q ( x ) - 2$ for some polynomial $q ( x )$.
E $\quad p ( x ) = ( x - 2 ) q ( x ) + 1$ for some polynomial $q ( x )$. F $\quad p ( x ) = ( x + 2 ) q ( x ) + 1$ for some polynomial $q ( x )$. G $\quad p ( x ) = ( x - 2 ) q ( x ) - 1$ for some polynomial $q ( x )$. H $\quad p ( x ) = ( x + 2 ) q ( x ) - 1$ for some polynomial $q ( x )$.

tmua 2016 Q2 1 marks Factorization and Root Analysis View

The expression $3 x ^ { 3 } + 13 x ^ { 2 } + 8 x + a$, where $a$ is a constant, has ( $x + 2$ ) as a factor. Which one of the following is a complete factorisation of the expression?
A $( x + 2 ) ( x - 1 ) ( 3 x - 2 )$ B $( x + 2 ) ( x + 1 ) ( 3 x - 2 )$ C $( x + 2 ) ( x + 1 ) ( 3 x + 2 )$ D $( x + 2 ) ( x - 3 ) ( 3 x + 2 )$ E $( x + 2 ) ( x + 3 ) ( 3 x - 2 )$ F $( x + 2 ) ( x + 3 ) ( 3 x + 2 )$

tmua 2017 Q4 1 marks Remainder by Linear Divisor View

When $\left( 3 x ^ { 2 } + 8 x - 3 \right)$ is multiplied by $( p x - 1 )$ and the resulting product is divided by $( x + 1 )$, the remainder is 24 .
What is the value of $p$ ?
A - 4
B 2
C 4
D $\frac { 8 } { 7 }$
E $\frac { 11 } { 4 }$

tmua 2018 Q5 1 marks Remainder by Linear Divisor View

The function f is defined by $\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$.
$a , b$ and $c$ take the values 1,2 and 3 with no two of them being equal and not necessarily in this order.
The remainder when $\mathrm { f } ( x )$ is divided by ( $x + 2$ ) is $R$.
The remainder when $\mathrm { f } ( x )$ is divided by ( $x + 3$ ) is $S$.
What is the largest possible value of $R - S$ ?
A - 26
B 5
C 7
D 17
E 29

tmua 2019 Q2 1 marks Divisibility and Factor Determination View

( 2 x + 1 )$ and $( x - 2 )$ are factors of $2 x ^ { 3 } + p x ^ { 2 } + q$.
What is the value of $2 p + q$ ?

tmua 2020 Q2 1 marks Divisibility and Factor Determination View

$(2x+1)$ and $(x-2)$ are factors of $2x^3 + px^2 + q$
What is the value of $2p + q$?
A $-10$
B $-\frac{38}{5}$
C $-\frac{22}{3}$
D $\frac{22}{3}$
E $\frac{38}{5}$
F $10$

tmua 2023 Q19 1 marks True/False or Multiple-Statement Evaluation View

In this question, $f ( x )$ is a non-constant polynomial, and $g ( x ) = x f ^ { \prime } ( x )$ $f ( x ) = 0$ for exactly $M$ real values of $x$. $g ( x ) = 0$ for exactly $N$ real values of $x$. Which of the following statements is/are true? I It is possible that $M < N$ II It is possible that $M = N$ III It is possible that $M > N$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

todai-math 2023 Q5 Proof of Polynomial Divisibility or Identity View

5 Go to the solutions page
Consider the polynomial $f(x) = (x-1)^2(x-2)$.

[(1)] Let $g(x)$ be a polynomial with real coefficients, and let $r(x)$ be the remainder when $g(x)$ is divided by $f(x)$. Show that the remainder when $g(x)^7$ is divided by $f(x)$ equals the remainder when $r(x)^7$ is divided by $f(x)$.
[(2)] Let $a$, $b$ be real numbers, and let $h(x) = x^2 + ax + b$. Let $h_1(x)$ be the remainder when $h(x)^7$ is divided by $f(x)$, and let $h_2(x)$ be the remainder when $h_1(x)^7$ is divided by $f(x)$. Find all pairs $a$, $b$ such that $h_2(x)$ equals $h(x)$.

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turkey-yks 2010 Q9 Divisibility and Factor Determination View

$$P(x) = 2x^{3} - (m+1)x^{2} - nx + 3m - 1$$
Given that the polynomial is completely divisible by $x^{2} - x$, what is $m - n$?
A) $\frac{-1}{3}$
B) $\frac{-1}{2}$
C) $\frac{3}{2}$
D) $2$
E) $3$

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12