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The Polynomial Remainder Theorem 2019-02-23
From pasandi:
f(x) is a quadratic polynomial. when f(x) is divided by (x-1),(x-2) & (x+2) the remainders respectively are -1, 4 and 2 how to find the f(x) in a question like this?
Answered by Penny Nom.
The degree of 2x + 7 2016-10-17
From Deejay:
How about when the given has no degree and the other has a constant only, For example 2x + 7, where's the leading term and what is its leading coefficient?
Answered by Penny Nom.
The remainder theorem 2016-03-27
From Pratyasha:
A quadratic polynomial when divide by x+2 leaves a remainder of 1 and when divided by x-1 leaves a remainder of 4. What will be the remainder if it is divided by (x+2)(x-1)?
Answered by Penny Nom.
Polynomial division 2016-03-25
From Ashley:

The instructions say to "perform the division".

Answered by Penny Nom.
3 over "x" plus 5 2015-02-23
From Kenny:
3 over "x" plus 5. I don't know this is either a polynomial and if so how would you define it.
Answered by Harley Weston.
The degree of a polynomial 2015-01-05
From Nichole:
How do I determine the degree of polynomials? I've searched this on sites but they are all so confusing! Is there a simple explanation or way to find what the degree is? Some examples are: 6x^4 10x^2yz^5 and 3m^2n^7-10m^8. I also have to say that I am under the impression that this symbol ^ means the number after it is an exponent.
Answered by Robert Dawson and Harley Weston.
Two quadratic polynomials 2014-10-28
From khushboo:
The zeroes of a quadratic polynomial x^2+ax+b are 'c' and 'd' and the zeroes of a quadratic polynomial x^2+cx+d=0 are 'a' and 'b'. find the numerical value of a+b+c+d where a, b, c and d are non zero integers.
Answered by Penny Nom.
We can't write sinx and cosx as a finite polynomial. 2013-03-31
From rimoshika:
prove that we can't write sinx and cosx as a finite polynomial.
Answered by Walter Whiteley.
Solve 2x^4+x^3-37x>26x^2+12 using a graphical approach. 2012-12-29
From Tehmas:
Solve 2x^4+x^3-37x>26x^2+12 using a graphical approach.
I don't know how to find the x-intercepts or y-intercepts in this form.

Answered by Penny Nom.
Two polynomials are divisible by x - 2 2012-12-26
From Tehmas:
Determine the values of m and n so that the polynomials 2x^3+mx^2+nx-3 and x^3-3mx^2+2nx+4 are both divisible by x-2.
Answered by Penny Nom.
Multiplying binomials 2012-05-14
From Jordan:
How do I solve the equation:

Answered by Penny Nom.
A quartic polynomial that is a perfect square 2012-02-05
From archit:
If P(x)=x^4+ax^3+bx^2-8x+1 is a perfect square then (a+b)=?
Answered by Penny Nom.
Factoring with fractions 2011-02-15
From Megan:
Hi there, I'm working on factoring polynomials but this question has me quite puzzled.. im a college student in my first year.


Answered by Penny Nom.
Identify each polynomial by its degree and number of terms 2011-01-10
From betty:
Write in standard form and identify each polynomial by its degree and number of terms. How do you do this?
Answered by Penny Nom.
A polynomial remainder 2010-12-25
From Asterdine:
When a polynomial f(x) is divided by x+1 and x+2, the remainders are 3 and 7 respectively. Find the remainder when f(x) is divided by (x+2)(x+1).
Answered by Penny Nom.
z^5 - 3z^4 + 2z^3 + z^2 - 3z + 2 2010-11-06
From Kumar:
would you please solve this problem, related to complex numbers.

Find all the roots of :

z^5 - 3z^4 + 2z^3 + z^2 - 3z + 2

Answered by Robert Dawson and Penny Nom.
A fourth degree polynomial function with real coefficients 2010-10-30
From Ryan:
Question from Ryan, a student: What is the fourth degree polynomial function with real coefficients that has -1,1 and I as zeros and such that f(x)=160
Answered by Harley Weston.
A polynomial with integer coefficients 2010-10-23
From Adori:
Question from Adori, a student: Find a polynomial with integer coefficients that has the given zeros: 2/3, 4, root of 3i, root of -3i.
Answered by Harley Weston.
A Taylor polynomial for (lnx)/x 2010-09-29
From Dave:
I have a series problem that I cannot solve. The problem asks for you to compute a Taylor polynomial Tn(x) for f(x) = (lnx)/x. I calculated this poly out to T5(x) and attempted to use this to identify a pattern and create a series in order to calculate Tn(x). However, the coefficients on the numerator out to F5prime(x) are as follows: 1, -3, 11, -50, 274... Ok, so the negative is an easy fix -> (-1)^n-1. But the other coefficients are stumping me. I can't see any sort of pattern there and I've tried every trick I know. Is there another way to go about this? Thanks!
Answered by Chris Fisher.
Distribution with Fractional Terms 2010-06-11
From Vincent:
Compounded semiannually. P dollars is invested at annual interest rate r for 1 year. If the interest iscompounded semiannually, then the polynomial P(1 + r/2)^2 represents the value of the interest after 1 year. Rewrite this expression without parentheses. Evaluate the polynomial if P=$200 and r = 10%.
Answered by Janice Cotcher.
Rational root theorem 2010-04-14
From deciree:
How could i get the value of x with this equation: 1/13x^3+12x/13=1/2 Thank you!
Answered by Harley Weston.
Factors and zeros of a polynoial 2010-03-17
From Susan:
find a polynomial function of degree three with -3 as a zero of multiplicity 2 and 4 as a zero of multiplicity 1.
Answered by Tyler Wood.
Zeros of a polynomial 2010-03-01
From Gavin:
Suppose the polynomial R(x) = a_9x^9+a_8x^8+...+a_1x+a_0 has real coefficients with a_9≠0. Suppose also that R(x) has the following zeros:

Using this info, answer the following.

a. What is another zero of R(x)?
b. At most how many real zeros of R(x) are there?
c. At most how many imaginary zeros of R(x) are there?

p.s. I used _ for subscript
thanks so much

Answered by Harley Weston.
Four dots are randomly placed on an 8x8 grid 2010-02-26
From bobbym:
Four dots are randomly placed on an 8x8 grid, compute the probability that no row or column contains more than one dot.
Answered by Claude Tardif.
(x+3) is a factor of 2x^3-3x^2+px+30 2010-01-09
From Ruby:
(x+3) is a factor of 2x^3-3x^2+px+30.
Determine the value of p

Answered by Penny Nom.
Coefficients 2009-11-17
From Da'mon:
The Coefficients of the expression 8 - 5x - 4 + 3x are 5 and 3 Whether the statement is true or false.
Answered by Penny Nom.
3x^3-3x-3 and 3x+6 2009-11-04
From spencer:
My question is 3x^3-3x-3 devided by 3x+6
Answered by Penny Nom.
The roots of a polynomial 2009-11-03
From Jess:
Find the value of a positive integer k such that f(x)=x^2 - 2kx + 55 has roots at k +3 and k - 3.
Answered by Penny Nom.
The zeros of a polynomial 2009-09-25
From jordan:
find a polynomial of lowest degree with only real coefficients and having the given zeros. 1+2i, 2 (multiplicity 2)
Answered by Penny Nom.
Fourth and fifth degree polynomials 2009-08-20
From Evin:
hello.i am a student . ax^4+bx^3+cx^2+dx+e=0 x=? i want to learn the solution or formula of equations of the fourth and fifth degree...PLEASE
Answered by Robert Dawson.
Using the Remainder Theorem to Find an Unknown Coefficient. 2009-07-06
From Mukulu:
What is the value of a if 2x2-x-6 ,3x2-8x+4 and ax3-10x-4 have a common factor.
Answered by Janice Cotcher.
Multiplication of polynomials 2009-04-07
From Carla:
I am struggling to understand Multiplication of Polynomials. No matter how hard I try to understand Multiplication of Polynomials, I just can't get it!

The problem that I am trying to solve is this :

-3x^3y(-y + 2 -x^2 + x)

Answered by Robert Dawson.
Leading terms and leading coefficients 2009-03-31
From Susan:
Write the polynomial P(x)=x^3-3x^4+17x+11-4/3x^2 in descending order. Identify the leading term and the leading coefficient.
Answered by Penny Nom.
Find a polynomial function with the indicated zeros and satisfying the given conditions 2009-03-23
From Kristen:
Find a polynomial function with the indicated zeros and satisfying the given conditions. Simplofy your answer (no imaginary numbers or parentheses in the answer) Zeros: 1+2i, 1-2i, 5 ; f(-2)=1
Answered by Harley Weston.
A cubic polynomial 2009-03-11
From Pedro:
I am having trouble figuring out the soultions for the polynomial equation


Answered by Robert Dawson.
Primes and polynomials 2008-11-24
From Andrew:
How effective are the two
pairs of polynomials, taken together, at "excluding" any prime p.

1. (a^2 - a.b + b^2) and (a^2 + a.b + b^2) Where b > a > p and (a, b, p) = 1

2. (c^2 - c.a + a^2) and (c^2 - c.b + b^2) Where c > b > a > p and (a, b, c, p) = 1
Answered by Victoria West.

A question about primes and polynomials 2008-11-16
From Andrew:
Is it possible for (a^2 - a.b + b^2) to be divisible by a prime p where b > a > p and a, b, and p are relatively prime? I let a = n.p + x and b = m.p + y (where both x and y are < p) and after dividing throughout by p one gets to decide whether (x^2 - x.y + y^2) ==0 mod(p) ?! It seems unlikely unless p = (x^2 - x.y + y^2)? (Note:- All letters represent positive integers.)
Answered by Chris Fisher and Victoria West.
Factoring x^2 + 729 2008-08-19
From peter:
hello I,am having trouble factorising a polynomial into polynomial factors with real coefficients please can you help the polynomial is x^2+729
Answered by Harley Weston.
Polynomial division and remainders 2008-07-02
From RAM:
If the polynomial x4-6x3+16x2-25x+10 is divided by another polynomial x2-2x+k, the remainder comes out to be x+a. Find k and a.
Answered by Janet Cotcher and Harley Weston.
4x^3 - 6x + 2 = 0 2008-05-28
From paul:
4x^3 - 6x + 2 = 0
Answered by Janice Cotcher.
A root of a polynomial 2008-05-03
From john:
here is a word problem that my teacher gave us to do can you help me understand it

Give possible formulas (in factored form) and sketch the graphs for two different polynomials that have roots -3, -1, and 2, but cross the x-axis only at x=2 and have y-intercept 4.5. What is the lowest degree such a polynomial can have? Can it have an even degree? An odd degree? Explain why.

her term "root" has completely thrown me off, i've reviewed my book and cant find what she is referring to. can anyone help?

Answered by Penny Nom.
For what values of a does x^3-13x+a have 3 distinct roots? 2008-04-25
From Jyotishman:
What can be the value of a for the equation to have 3 roots?

Answered by Harley Weston.
A polynomial equation 2008-04-18
From Joan:
Find all polynomials P(x) satisfying x^2P(x) + P(1-x) = 2x-x^4
Answered by Harley Weston.
A complex cubic polynomial 2008-03-26
From wael:
how do we solve in C the following equation:
z^3 + (5i-6)z^2 + (9-24i)z + 13i + 18 = 0 if it admits a pure imaginary root?

Answered by Harley Weston.
5x^2 - 45 2008-02-11
From Tiana:
factor: 5x^2 - 45
Answered by Stephen La Rocque.
Explaining the factoring for the difference of cubes 2008-01-16
From Bill:
A student asked me where did the "difference of cubes" and "sum of cubes" come from. I did not have an answer for her. She is very bright and understands how they work but wanted to know where they derived from. Any help you can offer would be great. Thanks
Answered by Stephen La Rocque.
Imaginary roots 2007-12-09
From Josh:
What is the correlation between imaginary roots (of a quadratic or other polynomial equation) and the graph of the equation? As in, how can one represent imaginary solutions graphically (and why does that work)?
Answered by Harley Weston.
6(x-1)-3(x+1)-12=56+5x 2007-10-24
From Megan:
Answered by Penny Nom.
(5C squared + 14C + 11) divided by (5C + 4) 2007-09-20
From John:
(5C squared + 14C + 11) divided by (5C + 4)
Answered by Harley Weston.
What is the inverse function of y=x^3 +x? 2007-09-19
From saman:
what is the inverse function of y=x^3 +x
Answered by Harley Weston.
16x^2+4xy^2+8x divided by 4xy 2007-09-02
From Ricardo:
how can i solve that problem of polynomial division?

16x^2+4xy^2+8x divided by 4xy

Answered by Penny Nom.
(x^3)-(4x^2)+(4x)=0 2007-08-15
From Amanda:

Answered by Leeanne Boehm.
Cubic polynomials 2007-08-14
From Jalon:
The cubic function f(x) = (x+2)^3 touches the x axis only once at x = -2 (negative two). it could also be written as f(x) = (x+2)(x+2)(x+2)

Investigate the cubic functions below as well as the one above and comment clearlt and fully on where they touch/intersect the x axis, and how these points relate to the given function.

a) f(x) = (x – 3)(x +4)(x – 2)
b) f(x) = x(x + 1)^2 (^2 = that means it squared)

Your comments should reference graphs illustrating your conclusions and display another 2 trinomial graphs that demonstrate your conclusions. particular attention should be given to the number of times your function crosses or touches the X axis. i dont know how this works but could you send the answer to..... spongy_91@hotmail.com

Answered by Stephen La Rocque.
A polynomial equation 2007-07-28
From claire:
Hi I sent this question a while ago but I think I put in the wrong email address. So here it goes again....

If 4x^3 - 6x^2 + 1 = (x-2)(x+1)Q(x) +ax+ b where Q(x) is a polynomial, find a and b.

Answered by Penny Nom.
A matrix of polynomials 2007-07-18
From Mac:
can you please help me out to solve this ?

Let A be a n*n matrix, the elements of which are real (or complex) polynomial in x. If r rows of the determinant becomes identical when x=a, then the determinant
A) has a factor of order r
B) has a factor or order > r
C) has no factor
D) has a factor of order < r

Answered by Harley Weston.
A polynomial with integer coefficients 2007-06-20
From Krys:
How do I find a polynomial P(x) with integer coefficient that has degree 4 and zeros 1-2i and 1 (with multiplicity of 2)?
Answered by Penny Nom.
Simplifying a quartic rational expression using long division 2007-06-14
From Megan:
Answered by Stephen La Rocque and Penny Nom.
Rational polynomial expressions 2007-06-09
From a student:
I have a question that continues to lead me to the answer x/(x - 1) but according to my math book, the answer should be 1/(x - 1). Who is right? Who is wrong? Here is the question: (x^2 - 6x - 27)/(x^2 - 11x + 18) DIVIDED by (x^3 + 2x^2 - 3x)/(x^2 - 2x)
Answered by Stephen La Rocque.
The lcm of polynomials 2007-05-31
From devin:
x^3-8 , x^3-4, x^2+4x+4
Answered by Penny Nom.
Factoring a trinomial in two variables 2007-05-23
From Ema:
How do you factor this completely. x^5y^3-3x^4y^2-28x^3y
Answered by Stephen La Rocque.
Division of polynomials 2007-04-14
From Rajendra:
Find the remainder when x3+3x+1 is divided by x+pi
Answered by Penny Nom.
Factoring polynomials 2007-02-14
From Joe:
I am in the eighth grade, and we are learning the equivalent of Algebra 2. I have no ides how to factor (x-2)(x^2-1)-6x-6 You help is most aprreciated. Thank you! Joe
Answered by Stephen La Rocque.
Multiplication of polynomial 2007-02-09
From John:
hello.. i just want to ask how can i solve this?? (2x+3)(6x+8)
Answered by Steve La Rocque, Penny Nom and Brennan Yaremko.
The product of polynomials 2006-07-13
From Lynne:
If you multiply two polynomials, what can you say about the degree of their product? What exponent applies here?
Answered by Stephen La Rocque.
Adding polynomials and counting the terms 2006-04-28
From Kris:
Suppose that two polynomials, each containing 3 terms, are added. Is it possible for the sum to contain more than 3 terms? fewer than 3 terms? exactly 3 terms? Explain.
Answered by Leeanne Boehm and Steve La Rocque.
x^2+2x-3 is a factor f(x)=x^4+2x^3-7x^2+ax+b 2006-04-06
From Maha:
Given that x2+2x-3 is a factor f(x)=x4+2x3-7x2+ax+b, find a and b and hence factor f(x) completely.
Answered by Penny Nom.
A fifth degree polynomial 2006-03-13
From Forrest:
When f(x) = x5 - 3x4 - 6x3 + 3ax2 - 24 is divided by x2 - 2 the remainder is bx.
Find the values of a and b.
Hence solve the equation f(x) = -8x

Answered by Penny Nom.
f(x) is a polynomial of degree 3. 2006-03-13
From Meadow:
f(x) is a polynomial of degree 3. It leaves a remainder of 10 and 4 when divided by x + 1 and x - 2 respectively. Given also that f(1) = f(-2) = 0, find the remainder when f(x) is divided by x - 3.
Answered by Penny Nom.
Factors of a cubic 2006-03-08
From Skye:
Show that (x+2) is a factor of f(x) where:

f(x) = px3 + (4p+2)x2 + (5p+4)x + 2p

Find the range of values for which the equation f(x) = 0 has 3 distinct real roots

Answered by Penny Nom.
X^2 + 4X + 6 is a factor of X^4 + rX^2 + s 2006-02-11
From Katie:
If X2 + 4X + 6 is a factor of X4 + rX2 + s, then find the value of r+s?
Answered by Steve La Rocque and Penny Nom.
Degree 2006-02-09
From Jessica:
I am a 9th grade home school student and have started doing degrees of terms in my math book. The following is some examples they give:

25a to the 4th power is a 4th degree term

67c to the 9th power is a 9th degree term

x is really x to the 1st power so it is a 1st degree term

10 is really 10x to the 0 power so it is a 0 degree term

They go on to say that every constant is a zero degree term.

My question is why isn't a constant, like 10, simply to the 1st power (making is a 1st degree term) like x.

Answered by Claude Tardif.
A graph with certain properties 2004-11-22
From A student:
i was asked as a question in coursework to sketch the graph with the following characteristics:
a double root at -3
a pair of imaginary roots
an x-intercept at 6
a root at 4 which is not a double root

Answered by Penny Nom.
Imaginary roots of a ploynomial 2004-10-31
From Jennifer:
how to find the roots of a polynomial equation if it would be imaginary?
Answered by Harley Weston.
Polynomials 2004-01-25
From Bruce:

A polynomial is defined as
Polynomial functions are functions that have this form:
f(x) = anxn + an-1xn-1 + ... + a1x + a0
The value of n must be an nonnegative integer. That is, it must be whole number; it is equal to zero or a positive integer.
The coefficients, as they are called, are an, an-1, ..., a1, a0. These are real numbers.


  1. why must n be positive?
  2. what are some historical facts about the evolution of the definition?

Answered by Harley Weston.
Polynomial division 2004-01-13
From Efrat:

How do we divide the following polynomials?

(x3 + x2 - x + 2) / (x + 2)

Answered by Penny Nom.
Polynomials degree 4 and bigger 2003-06-26
From Stephen:

How do I show kids how to find all the zeros for polynomials degree 4 and bigger.

For examples:
r(x) = x5-11x3-7x2+77 = (x2-11)(x3-7) and s(x) = x4-121 = (x2-11)(x2+11)

Also am I correct when I say that the following are irreducible?

f(x) = x2-11
g(x) = x3-7
h(x) = x4-5

Answered by Harley Weston.
The HCF and LCM of polynomials 2003-05-20
From Charanpal:

Find the HCF and LCM of the polynomials given below. Verify that he productof these HCF and LCM differs from the product of the polynomials, if at all, by a factor of -1

  1. 1-x2 and x3 -1
  2. 1 - x2 and x4 - 1

Answered by Penny Nom.
The factor theorem 2002-09-10
From Yael:
Prove (x - a - b) is a factor of x3 - a3 - b3 - 3ab (a + b)
Answered by Penny Nom.
A polynomial 2002-06-05
From Melissa:
I'd like to know what is a polynomial( the definition and an explication)? And is 7x a polynomial? and why?
Answered by Penny Nom.
A polynomial 2002-04-14
From A student:
A polynomial function is described by the following conditions: f(x) has two real zeros at x= -2 and x= 1, each of multiplicity 2 f(x) has two complex zeros as x goes to -infinity, f(x) goes to +infinity as x goes to +infinity, f(x) goes to +infinity f(x) has a y-intercept at (0,2)

Sketch a polynomial function that satisfies the above conditions.

Write an equation for function f(x)

Write another function g(x) that also satisfies the above parameters.

Answered by Penny Nom.
A degree 5 polynomial 2002-02-17
From Carla:
Write an equation for a polynomial of degree 5 given the following zeros:
-7+3i, -2+sqrt(13), 6

Answered by Penny Nom.
Dividing polynomials 2001-02-19
From Janna:
I have two questions involving dividing polynomials by polynomials.

Here's the first one:

Two factors of 12a4 -39a2 + 8a - 8a3 + 12 are a - 2 and 2a + 1. Find the other factors. The other question I'm stuck on is: When 10x3 + mx2 - x + 10 is divided by 5x-3, the quotient is 2x2 + nx - 2 and the remainder is 4. Find the values of m and n.

Answered by Penny Nom.
Polynomials and exponents 2001-01-15
From A student:
I am duing a project in math on polynomials and exponents. I need a real life usage of polynomials and exponents for my project.
Answered by Penny Nom.
Subtracting polynomials 2000-05-13
From Teresa:
I am suppose to subtract the lower polynomial from the one above it.
-3a- 6b+8f

Answered by Penny Nom.
polynomial functions 1999-11-19
From Quinn:
  1. Without fully factoring the following show that they all have the same zeros:

    I: x4-x2+2x+6
    II: -x4+x2-2x-6
    III: -4x4+4x2-8x-24
    IV: 10x4-10x2+20x+60

  2. When P(x)=x3-3x2+5x+1 and G(x)=x3-2x2-x+10 are each divided by(x-a) the remainders are equal. At what coordinate point does the graph of P(x) intersect G(x)?

Answered by Walter Whiteley.
The three cousins 1999-06-01
From Irena:
I have three cousins on my dad's side. The product of their ages is 84. Next year the youngest will be 1/4 the age of the middle cousin. In 2 years from now, the oldest will be twice the current age of the middle. What is the age of my oldest cousin?
Answered by Harley Weston.
Question about 3rd degree polynomials 1999-04-23
From Patrick Bryan:
What is the general solution to the equation with the form:

a*x^3 + b*x^2 + c*x + d = 0

I have once seen a solution to this a few years ago, but I do not recall if it was a general solution. What I do know, is that you could simplify this equation to:

a*x'^3 + p*x' + q = 0...
Answered by Doug Farenick.

Factoring polynomials 1999-03-29
From Karen:
I have been unable to factorise a polynomial equation and was wondering if you could please help. It is level (10-12) maths. The polynomial is x3 + x2 - 24x + 36
I have tried a few factorisation methods such as foctorisation by grouping but it won't work this polynomial. Please help.

Answered by Jeff Walters and Jack LeSage.
The Factor Theorem 1998-01-18
From Dhruv Bansal:
Hi, I'm a ninth grade student doing a math project for school. I'm trying to learn about various ways of solving 3rd and 4th degree polynomial equations using the remainder theorem, the factor theorem, and synthetic division. The books I have all mention x - r, which I know nothing about, not even how to get it.

I would really appreciate any information on this. Thanks.

Dhruv Bansal
Answered by Penny Nom.

Polynomials 1998-01-13
From Sarah Storkey:
Hi, My name is Sarah, My question is at the junior level.

My question is, what is a polynomial?
Answered by Harley Weston.

Polynomials 1997-10-07
From Sheryl and Jeff:
I'm a math teacher in Jerusalem, Israel.

I'm teaching about graphing polynomial functions in a pre-calc class. A student asked me what they're good for. I couldn't give her a good example. Do you have one. Thanks.
Answered by Penny Nom.

Roots of a Polynomial 1996-12-06
From Paula Miller:
Why is the solution called a "root"? Why not just the "x-intercepts"?
Answered by Chris Fisher and Harley Weston.
Un polynome de degré 3 2010-10-09
From ait:
comment determiner n polynome P de degré 3 verifiant pour tout reel x la relation p(x+1)-p(x)=x˛
Answered by Claude Tardif.



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