







The Polynomial Remainder Theorem 
20190223 

From pasandi: f(x) is a quadratic polynomial. when f(x) is divided by
(x1),(x2) & (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 
20161017 

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 
20160327 

From Pratyasha: A quadratic polynomial when divide by x+2 leaves a remainder of 1 and when divided by x1 leaves a remainder of 4. What will be the remainder if it is divided by (x+2)(x1)? Answered by Penny Nom. 





Polynomial division 
20160325 

From Ashley: (4x4+6x^3x^2)/(3x2)
The instructions say to "perform the division". Answered by Penny Nom. 





3 over "x" plus 5 
20150223 

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 
20150105 

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^710m^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 
20141028 

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. 
20130331 

From rimoshika: prove that we can't write sinx and cosx as a finite polynomial. Answered by Walter Whiteley. 





Solve 2x^4+x^337x>26x^2+12 using a graphical approach. 
20121229 

From Tehmas: Solve 2x^4+x^337x>26x^2+12 using a graphical approach.
I don't know how to find the xintercepts or yintercepts in this form. Answered by Penny Nom. 





Two polynomials are divisible by x  2 
20121226 

From Tehmas: Determine the values of m and n so that the polynomials 2x^3+mx^2+nx3 and x^33mx^2+2nx+4 are both divisible by x2. Answered by Penny Nom. 





Multiplying binomials 
20120514 

From Jordan: How do I solve the equation:
3(x+4)(x6) Answered by Penny Nom. 





A quartic polynomial that is a perfect square 
20120205 

From archit: If P(x)=x^4+ax^3+bx^28x+1 is a perfect square then (a+b)=? Answered by Penny Nom. 





Factoring with fractions 
20110215 

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.
(x^2a^2/xy)(xy/x+a) Answered by Penny Nom. 





Identify each polynomial by its degree and number of terms 
20110110 

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 
20101225 

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 
20101106 

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 
20101030 

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 
20101023 

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 
20100929 

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)^n1. 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 
20100611 

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 
20100414 

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 
20100317 

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 
20100301 

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:
2,
3,
i
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 
20100226 

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^33x^2+px+30 
20100109 

From Ruby: (x+3) is a factor of 2x^33x^2+px+30.
Determine the value of p Answered by Penny Nom. 





Coefficients 
20091117 

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^33x3 and 3x+6 
20091104 

From spencer: My question is 3x^33x3 devided by 3x+6 Answered by Penny Nom. 





The roots of a polynomial 
20091103 

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 
20090925 

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 
20090820 

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. 
20090706 

From Mukulu: What is the value of a if 2x2x6 ,3x28x+4 and ax310x4 have a common factor. Answered by Janice Cotcher. 





Multiplication of polynomials 
20090407 

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 
20090331 

From Susan: Write the polynomial P(x)=x^33x^4+17x+114/3x^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 
20090323 

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, 12i, 5 ; f(2)=1 Answered by Harley Weston. 





A cubic polynomial 
20090311 

From Pedro: I am having trouble figuring out the soultions for the polynomial equation
6x^317x^2+4x+12=0 Answered by Robert Dawson. 





Primes and polynomials 
20081124 

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 
20081116 

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 
20080819 

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 
20080702 

From RAM: If the polynomial x46x3+16x225x+10 is divided by another polynomial x22x+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 
20080528 

From paul: 4x^3  6x + 2 = 0 Answered by Janice Cotcher. 





A root of a polynomial 
20080503 

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 xaxis only at x=2 and have yintercept 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^313x+a have 3 distinct roots? 
20080425 

From Jyotishman: What can be the value of a for the equation to have 3 roots?
x^313x+a=0 Answered by Harley Weston. 





A polynomial equation 
20080418 

From Joan: Find all polynomials P(x) satisfying x^2P(x) + P(1x) = 2xx^4 Answered by Harley Weston. 





A complex cubic polynomial 
20080326 

From wael: how do we solve in C the following equation:
z^3 + (5i6)z^2 + (924i)z + 13i + 18 = 0 if it admits a pure imaginary root? Answered by Harley Weston. 





5x^2  45 
20080211 

From Tiana: factor:
5x^2  45 Answered by Stephen La Rocque. 





Explaining the factoring for the difference of cubes 
20080116 

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 
20071209 

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(x1)3(x+1)12=56+5x 
20071024 

From Megan: 6(x1)3(x+1)12=56+5x Answered by Penny Nom. 





(5C squared + 14C + 11) divided by (5C + 4) 
20070920 

From John: (5C squared + 14C + 11) divided by (5C + 4) Answered by Harley Weston. 





What is the inverse function of y=x^3 +x? 
20070919 

From saman: what is the inverse function of y=x^3 +x Answered by Harley Weston. 





16x^2+4xy^2+8x divided by 4xy 
20070902 

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 
20070815 

From Amanda: solve.
(x^3)(4x^2)+(4x)=0 Answered by Leeanne Boehm. 





Cubic polynomials 
20070814 

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 
20070728 

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 = (x2)(x+1)Q(x) +ax+ b where Q(x) is a polynomial, find a and b. Answered by Penny Nom. 





A matrix of polynomials 
20070718 

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 
20070620 

From Krys: How do I find a polynomial P(x) with integer coefficient that has degree 4 and zeros 12i and 1 (with multiplicity of 2)? Answered by Penny Nom. 





Simplifying a quartic rational expression using long division 
20070614 

From Megan: x+2/12x^4+17x^3+0x^2+8x40= Answered by Stephen La Rocque and Penny Nom. 





Rational polynomial expressions 
20070609 

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 
20070531 

From devin: x^38 , x^34, x^2+4x+4 Answered by Penny Nom. 





Factoring a trinomial in two variables 
20070523 

From Ema: How do you factor this completely. x^5y^33x^4y^228x^3y Answered by Stephen La Rocque. 





Division of polynomials 
20070414 

From Rajendra: Find the remainder when x3+3x+1 is divided by x+pi Answered by Penny Nom. 





Factoring polynomials 
20070214 

From Joe: I am in the eighth grade, and we are learning the equivalent of Algebra 2. I have no ides how to factor (x2)(x^21)6x6 You help is most aprreciated. Thank you! Joe Answered by Stephen La Rocque. 





Multiplication of polynomial 
20070209 

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 
20060713 

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 
20060428 

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+2x3 is a factor f(x)=x^4+2x^37x^2+ax+b 
20060406 

From Maha: Given that x^{2}+2x3 is a factor f(x)=x^{4}+2x^{3}7x^{2}+ax+b, find a and b and hence factor f(x) completely. Answered by Penny Nom. 





A fifth degree polynomial 
20060313 

From Forrest: When f(x) = x^{5}  3x^{4}  6x^{3} + 3ax^{2}  24 is divided by x^{2}  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. 
20060313 

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 
20060308 

From Skye: Show that (x+2) is a factor of f(x) where:
f(x) = px^{3} + (4p+2)x^{2} + (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 
20060211 

From Katie: If X^{2} + 4X + 6 is a factor of X^{4} + rX^{2} + s, then find the value of r+s? Answered by Steve La Rocque and Penny Nom. 





Degree 
20060209 

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 
20041122 

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 xintercept at 6
a root at 4 which is not a double root Answered by Penny Nom. 





Imaginary roots of a ploynomial 
20041031 

From Jennifer: how to find the roots of a polynomial equation if it would be imaginary? Answered by Harley Weston. 





Polynomials 
20040125 

From Bruce:
A polynomial is defined as
Polynomial functions are functions that have this form:
f(x) = a_{n}x^{n} + a_{n1}x^{n1} + ... + a_{1}x + a_{0}
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 a_{n}, a_{n1},
..., a_{1}, a_{0}. These
are real numbers.
Questions:
 why must n be positive?
 what are some historical facts
about the evolution of the definition?
Answered by Harley Weston. 





Polynomial division 
20040113 

From Efrat:
How do we divide the following polynomials?
(x^{3} + x^{2}  x + 2) / (x + 2)
Answered by Penny Nom. 





Polynomials degree 4 and bigger 
20030626 

From Stephen:
How do I show kids how to find all the zeros for polynomials degree 4 and bigger.
For examples: r(x) = x^{5}11x^{3}7x^{2}+77 = (x^{2}11)(x^{3}7) and s(x) = x^{4}121 = (x^{2}11)(x^{2}+11) Also am I correct when I say that the following are irreducible? f(x) = x^{2}11 g(x) = x^{3}7 h(x) = x^{4}5 Answered by Harley Weston. 





The HCF and LCM of polynomials 
20030520 

From Charanpal:
Question: 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  1x^{2} and x^{3} 1
 1  x^{2} and x^{4}  1
Answered by Penny Nom. 





The factor theorem 
20020910 

From Yael: Prove (x  a  b) is a factor of x^{3}  a^{3}  b^{3}  3ab (a + b) Answered by Penny Nom. 





A polynomial 
20020605 

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 
20020414 

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 yintercept 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 
20020217 

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 
20010219 

From Janna: I have two questions involving dividing polynomials by polynomials. Here's the first one: Two factors of 12a^{4} 39a^{2} + 8a  8a^{3} + 12 are a  2 and 2a + 1. Find the other factors. The other question I'm stuck on is: When 10x^{3} + mx^{2}  x + 10 is divided by 5x3, the quotient is 2x^{2} + nx  2 and the remainder is 4. Find the values of m and n. Answered by Penny Nom. 





Polynomials and exponents 
20010115 

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 
20000513 

From Teresa: I am suppose to subtract the lower polynomial from the one above it. 3a+11b+2f 3a 6b+8f ________________ Answered by Penny Nom. 





polynomial functions 
19991119 

From Quinn:
 Without fully factoring the following show that they all have the same zeros:
I: x^{4}x^{2}+2x+6 II: x^{4}+x^{2}2x6 III: 4x^{4}+4x^{2}8x24 IV: 10x^{4}10x^{2}+20x+60  When P(x)=x^{3}3x^{2}+5x+1 and G(x)=x^{3}2x^{2}x+10 are each divided by(xa) the remainders are equal. At what coordinate point does the graph of P(x) intersect G(x)?
Answered by Walter Whiteley. 





The three cousins 
19990601 

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 
19990423 

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 
19990329 

From Karen: I have been unable to factorise a polynomial equation and was wondering if you could please help. It is level (1012) maths. The polynomial is x^{3} + x^{2}  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 
19980118 

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 
19980113 

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 
19971007 

From Sheryl and Jeff: I'm a math teacher in Jerusalem, Israel. I'm teaching about graphing polynomial functions in a precalc 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 
19961206 

From Paula Miller: Why is the solution called a "root"? Why not just the "xintercepts"? Answered by Chris Fisher and Harley Weston. 





Un polynome de degré 3 
20101009 

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. 

