# Questions on Algebra: Polynomials, rational expressions and equations answered by real tutors!

Algebra ->  Polynomials-and-rational-expressions -> Questions on Algebra: Polynomials, rational expressions and equations answered by real tutors!      Log On

1.The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=4 and roots of multiplicity 1 at x=0 and x=−3 . It goes through the point (5,32) .
Find a formula for P(x) .

You can put this solution on YOUR website!
Root x=4 of multiplicity 2 means that P(x) has a factor (x-4)^2
Roots of multiplicity 1 at x=0 means P(x) has a factor x and at
x=-3 means a factor (x+3)
****************************************************************
P(x) = x * (x-4)^2 * (x+3)
P(x) = (x^2+3x) * (x-4)^2
P(x) = (x^2+3x) * (x^2 -8x +16)
P(x) = x^4 -5x^3 -8x^2 +48x
****************************************************************
now we use point (5,32)
32 = 5^4 -5(5^3) -8(5^2) +48(5)
32 = 625 -625 -200 +240
to make this work we have to subtract 8 from the right side of =, therefore
****************************************************************
P(x) = x^4 -5x^3 -8x^2 +48x -8 and then
32 = 32

Question 994719: expand(y+3r)^3

You can put this solution on YOUR website!
expand(y+3r)^3
------------
Use Pascal's triangle --> coeff's are 1 3 3 1
---
(y+3r)^3 = 1*y^3 + 3*y^2*(3r)^1 + 3*y*(3r)^2 + 1*(3r)^3
=

Question 994571: simplify (x+y)(x^2-xy+y^2)
You can put this solution on YOUR website!
simplify (x+y)(x^2-xy+y^2) = x^3+y^3
-----------
Cheers,
Stan H.

Question 994116: Factorize 5z^2_15z_21
You can put this solution on YOUR website!
.

Question 993226: two students took an exam one of them achieved 9 marks more than the other and his marks were 56% of the sum of their marks

You can put this solution on YOUR website!
a=b+9
and
a=0.56*(a+b)
=> 100a=56*(a+a-9)
=> a = 42
so, b = 33

Question 993883: how do you sketch the graph of a polynomial function that has 3 real zeros, and two imaginary zeros. Like how would you graph this polynomial function 3x^5-4x^4+2x^3-x^2-x-7. It has 3 real zeros and 2 imaginary,but I don't know how to graph that.

You can put this solution on YOUR website!
you have to find the roots.
the graph will cross or touch the x-axis at the real roots.
the graph will not cross or touch the x-axis at the imaginary roots.
if the root has an even multiplicity, the graph will touch the x-axis but not cross it.
if the root has an odd multiplicity,the graph will cross the x-axis.
in between the real roots, you just have to test selected points in order to be able to sketch the graph.

check this reference out as it explains how to sketch a graph and it also discusses even roots and odd roots.

the graph of your equation looks like this:

since it only crosses the x-axis once, it has only 1 real root.
the other 4 must be complex.
fyi - complex roots always come in pairs.
you can have 2 or 4, but you can't have 3.

lots of good stuff in there.

check out tutorials 38 and 39.

basically you are looking for the real roots.

there are all kinds of tests to determine what they are.

once you found those, if the number of roots is not equal to the degree of the equation, then the other roots must be complex.

Question 993841: Two landscapes must mow a rectangular lawn that measures 100 feet by 200 feet. Each wants to mow no more than half of the lawn. The first starts by mowing around the outside of the lawn. How wide a strip must the first landscaper mow on each side of the four sides in order to mow no more than half of the lawn? The mower has a 24 inch cut. Approximate the required number of trips around the lawn.
I have no idea how to set up an equation for this word problem.

You can put this solution on YOUR website!
Best to try to break the steps into each trip around for the first mower/landscaper.

24 inches is 2 feet.
Trip 1 mows leaves unmowed
Trip 2 leaves unmowed
Trip 3 leaves unmowed
Trip n leaves unmowed , according to the developing pattern.

HALF the lawn's area is .

How many trips around, n, will leave an unmowed area of 10000?

and this would hopefully show a solved value for n to be a whole number.

Question 993820:
You can put this solution on YOUR website!
.
Empty request.

Question 993818: Hi,
I would like to say thank you for your help in advanvced.
Can you please give an explanation of why..
ax + bx -2bx factors into
x(a-b)
Thank you,
Anthony

You can put this solution on YOUR website!
ax + bx -2bx =
ax - bx = x(a - b)
note that bx - 2bx = -bx

Question 993744: howhow do I multiply a number by 2
You can put this solution on YOUR website!
Very carefully.

Question 993637: write a polynomial function of least degree with integral coefficients that have the given zero 3i
You can put this solution on YOUR website!
write a polynomial function of least degree with integral coefficients that have the given zero 3i
----
If the coefficients are integral and 3i is a zero, -3i is also a zero.
------
Ans: y = (x-3i)(x+3i)
---
y = (x^2 - (3i)^2)]
---
y = x^2 + 9
-----------
Cheers,
Stan H.

Question 993574: Please help me solve this problem. It looks simple but I have no idea where to start or what to do?
Solve: x+6/x=5

Found 2 solutions by solver91311, ikleyn:
You can put this solution on YOUR website!

You really need to consider using parentheses to make certain it is clear what parts of your expressions go in the numerator and what parts go in the denominator. Assuming you actually meant:

Multiply both sides by x.

Multiply both sides by the reciprocal of the coefficient on the variable.

John

My calculator said it, I believe it, that settles it

You can put this solution on YOUR website!
Please help me solve this problem. It looks simple but I have no idea where to start or what to do?
Solve: x+6/x=5
-------------------------------------------------------------------------

Is it this equation?

+ = .

If so, multiply both sides by x. You will get

= .

The roots are 2 and 3.

You can apply the quadratic formula, or the Viete's theorem, or to factor the left side explicitly: = (x-2)*(x-3).

Answer. x = 2 and/or x = 3.

Question 993493: I am a 7th grade pre-algebra student transffering up to take algebra with the 8th graders. I have been trying this problem for about 30 minutes now and I still dont get it. Can you please help out
a triangle has a rational area of 20 ft. what possible combinations or irrational and rational for the base and height would give a rational value of the area.

You can put this solution on YOUR website!
Area is (1/2) bh=20 sq ft. Units will be feet for base and height.
bh=40
h=40/b
b=40/h
As I interpret the question, anything on the below graph where both are positive will give a rational value for the area. If they want possible combinations, you can use anything on this line, like 8 and 5 or 5 and 8 (all are interchangeable), (40/9) and 9, 20/9 and 18, sqrt (40) and sqrt (40), two irrational numbers, sqrt 80 and sqrt (5).

Question 993243: A skydriver jump out of a plane traveling 10,000ft above the ground how high is she after 10 seconds? After 15 seconds
You can put this solution on YOUR website!
t=time falling, in seconds
.
Distance fallen=(1/2)(acceleration of gravity)(time in seconds)^2
Distance=1/2(32ft/s^2)(t^2)
For 10 seconds:
Distance fallen=1/2(32ft/s^2)(10s)^2=(16ft/s^2)(100s^2)=1600 feet
Height=10000ft-1600ft=8400 feet
ANSWER 1: She is 8400 feet high after 10 seconds.
For 15 seconds:
Distance fallen=1/2(32ft/s^2)(15s)^2=(16ft/s^2)(225s^2)=3600 feet
Height=10000ft-3600ft=6400 feet
ANSWER 2: She is 6400 feet high after 15 seconds.

Question 993115: = ?
You can put this solution on YOUR website!
.
= ?
-----------------------------------------------

Hello,

has a remarkable factorization.

= - =     (we added and subtracted )   ----->

= - =     (now apply the formula for the difference of squares)   ----->

= .

Therefore,

= .

Question 993057: express in the form a+bi:

1+2i/ 1- square root of -64

Answer by Edwin McCravy(13211)   (Show Source):
You can put this solution on YOUR website!
1+2i/ 1- square root of -64


Change  to  to  to  to

Multiply by the conjugate of the denominator over itself
which we can do because that is just multiplying by 1.

Use FOIL on top and bottom:

Combine like terms and eliminate terms that have 0 sum:

Substitute -1 for i2

Combine like terms:

To express in terms of a+bi, break into two fractions:

Reduce the fractions:

Move the i factor in the second term from the numerator
to the right:

Edwin

Question 993065: 2x^2+16x+2 factor using AC method
Answer by Edwin McCravy(13211)   (Show Source):
You can put this solution on YOUR website!
2x²+16x+2

Don't start out trying to use the AC method first.
Always look for a common factor first. The AC method
will wait till later -- if it can be used at all.

First factor out the common factor 2:

2(x²+8x+1)

As it turns out it is not factorable any further.  So
the AC method will not work here. So the final factored
form is simply

2(x²+8x+1)

Edwin

Question 992993: please help me with Find a polynomial f(x) of degree 3 with real coefficients and the following zeros: 4, -1+i

First change the signs on (x-(-1+i) and (x+(-1+i).
giving me (x+1+i) and (x-1+i)
I then disturbed and is left with (x^2+2xi-2)
Then i disturbed the (x-4): (x^3+2ix^2-2x-4x^2-8xi+8)
That is when i get confused.
,

You can put this solution on YOUR website!
.
Wrong.

Your polynomial should be with real coefficients.

Then, having the root -1+i, it must have the conjugated complex number -1-i as the other root.

So the roots must be 4, -1+i and -1-i.

Hence, the polynomial is (x-4)*(x-(-1+i))*(x-(-1-i)).

You can transform it further, if you want.

Question 992868: Find polynomials q and r such that f(x)=q(x)g(x)+r(x) where the degree of r is strictly less than degree of g.
f(x)=x^4-3x^3+2x^2-x+1 and g(x)=x^2-3

You can put this solution on YOUR website!
Find polynomials q and r such that f(x)=q(x)g(x)+r(x) where the degree of r is strictly less than degree of g.
f(x)=x^4-3x^3+2x^2-x+1 and g(x)=x^2-3
---
Divid
f(x) by g(x) to find the quotient(q(x)) and the remainder(r(x)).
----
Ans:
q(x) = x^2 - 3x +8
r(x) = -x + 25
------
Cheers,
Stan H.
------------

Question 992839: How would I solve this problem? im supposed to divide a rational expression.
Q^2÷2q/6q ÷ q^2-4/3q^2

You can put this solution on YOUR website!
How would I solve this problem? im supposed to divide a rational expression.
Q^2÷2q/6q ÷ q^2-4/3q^2
------------------
Invert the divisor and multiply.
The divisor is the fraction on the right side.
==================
It's not clear, add some parentheses.

Question 992784: Dwayne's garden is triangle-shaped with two equal sides and a third side that is 4 ft more than the length of an equal side. If the perimeter is 49 ft, how long is each side?
Which equation represents this?

Found 2 solutions by stanbon, Cromlix:
You can put this solution on YOUR website!
Dwayne's garden is triangle-shaped with two equal sides and a third side that is 4 ft more than the length of an equal side. If the perimeter is 49 ft, how long is each side?
Which equation represents this?
----
Sketch a picture of this isosceles triangle: equal sides are "x"; base = (x+4).
Equation:
x + x + x+4 = 49
3x = 45
x = 15 (length of each of the equal sides)
x+4 = 19 (length of the base)
---------
Cheers,
Stan H.
---------------

You can put this solution on YOUR website!
Hi there,
Dwayne's garden is an isosceles triangle.
Two equal sides (and two equal angles)
Length of an equal side = 'x'
Length of other side(s) = x + 4
Perimeter = the 3 sides added together.
49 = x + x + (x + 4)
49 = x + x + x + 4
Collect like terms
3x = 49 - 4
3x = 45
x = 15.
Two equal sides = 15 ft
Third side = 19 ft
Hope this helps :-)

Question 992620: Find a polynomial function that has the given zeros
9,-5

You can put this solution on YOUR website!

a polynomial function that has the given zeros
and will be:

Question 992558: Find the domain and range and sketch the graph for the functions f and g where f(x) is given by the polynomial below and g(x)=|f(x)|. Also use the graph to solve the inequality f(x)<0.
f(x)=9+3x-2x^2

You can put this solution on YOUR website!
.
Find the domain and range and sketch the graph for the functions f and g where f(x) is given by the polynomial below and g(x)=|f(x)|.
Also use the graph to solve the inequality f(x)<0.
f(x)=9+3x-2x^2
----------------------------------------------------------------------------

The domain of the function  f  is the entire number line,  i.e.  the set of all real numbers.  (It is so for any polynomial).

The domain of the function  g  is the same entire number line,  i.e. the set of all real numbers.
The plots of  f  and  g  are shown in the Figure below.  The plot of  g  is in green.  Partly it is over-posed the plot of  f.

The roots of f are  3  and  .
f(x) < 0  in two semi-infinite intervals  (, )  and  (3, ).

Question 992561: Plot the graph of the rational function f. Include the asymptotes, x and y-intercepts and at least one intermediate point between these points.

f(x)=x−2+1/x−1

You can put this solution on YOUR website!
Plot the graph of the rational function f. Include the asymptotes, x and y-intercepts and at least one intermediate point between these points.
f(x)= (x−2) + [1/(x−1)]
------

-----------
Cheers,
Stan H.
-----------

Question 992556: Write the polynomial as a product of linear factors. Hint: The polynomial has at least one rational zero.

p(x)=x^3+2x^2+3x+6

You can put this solution on YOUR website!
Write the polynomial as a product of linear factors. Hint: The polynomial has at least one rational zero.
p(x)=x^3+2x^2+3x+6
------
P(x) = x^2(x+2)+3(x+2)
------
P(x) = (x+2)(x^2+3)
------
P(x) = (x+2)(x+isqrt(3))(x-isqrt(3))
-----------
Cheers,
Stan H.
-----------

Question 992537: How does this process change if the coefficient of the quadratic term is not either 1 or can be made to be 1 by removing the GCF? For example, consider a quadratic expression such as 14x2 + 29x - 15.
You can put this solution on YOUR website!
equation is 14x^2 + 29x - 15 = 0
multiply the coefficient of the x^2 term by the constant term to get:
14 * 15 = 210
look for factors of 210 that will be equal to 210 when multiplied together and will be equal to 29 when added together.
since the constant term is negative, the two factors will have opposite signs because a positive times a negative is equal to a negative.
looking at all the factors of 210, i find that 35 and 6 are the factors that i need.
this is because 35 * 6 = 210 and 35 - 6 = 29.
the positive factor is 35.
the negative factor is 6.
now you split the middle term of the equation as shown below:
14x^2 + 29x - 15 = 0 becomes:
14x^2 + 35x - 6x - 15 = 0

now group the first two terms and the last two terms together to get:
(14x^2 + 35x) - (6x + 15) = 0

note the change of sign in the second of these grouped factors.
grouping with minus sings can be tricky.
- 6x - 15 becomes - (6x + 15) when you group those last two factors together.
that's because - (6x + 15) is equivalent to -6x - 15 when you expand it.

now factor out the common terms of reach group to get:
7x * (2x + 5) - 3 * (2x + 5)

what you are looking for here is that one of the factors in the secondf set is common to one of the factors in the first set.

the common factor that we were looking for here is 2x + 5

now you can factor out that common factor to get:

7x * (2x + 5) - 3 * (2x + 5) becomes:

(7x - 3) * (2x + 5)

multiply them out and you will see that you get the original equation.

(7x - 3) * (2x + 5) =
7x * 2x = 14x^2
+ 7x * 5 = 35x
- 3 * 2x = -6x
- 3 * 5 = -15

combine like terms and you get:

14x^2 + 29x - 15.

that's your original equation so you're done.

this method is tricky but gets you to the right answer fairly quickly if you do it right.

same with the box method.

the other method is guess and check where you go through a ritual of trying dfifferent factors until you arrive at the right one.

if any of these methods prove difficult, then go to the quadratic formula.

that will be you the factors in all case, whether or not the quadratic equation is factorable or not, and whether or not the factors are real or not.

here's some references on factoring that you might find helpful.

http://www.regentsprep.org/regents/math/algtrig/atv1/revfactorgrouping.htm

all kinds of stuff on the web.
all you need to do is a search on "factoring quadratic equations" or "factoring quadratics using box method" or "factoring quadratic equations using split the middle term method.

there are even other methods like the indian method.

personally, if i can't factor it easily using any of these methods, i go right to the quadratic formula since that is the method of last resort.

Question 992529: If m= -3 then what is the value of -2M

You can put this solution on YOUR website!
If m=-3
Then,
-2m=-2*(-3)
=6
HOPE THIS HELPS?

Question 992351: I need to find the real and complex solution(s) and whether it has a multiplicity.
Problem: f(x) = x^4-7x^2+ 12
Note: I am able to solve similar equations if the exponents are sequential. However, I am unsure how to solve when they are not. Example...x^4 - X^3 + 25x^2 - 25x = Solutions are x =1, -5i, and 5i.
Many Thanks!

Found 3 solutions by MathTherapy, ikleyn, stanbon:
You can put this solution on YOUR website!

I need to find the real and complex solution(s) and whether it has a multiplicity.
Problem: f(x) = x^4-7x^2+ 12
Note: I am able to solve similar equations if the exponents are sequential. However, I am unsure how to solve when they are not. Example...x^4 - X^3 + 25x^2 - 25x = Solutions are x =1, -5i, and 5i.
Many Thanks!


------ Factoring out difference of 2 squares in
OR            OR



You can put this solution on YOUR website!
.
You need to find the roots of the equation

= .

Correct?
-----------------------------

Introduce new variable  y = .  Then the equation takes the form

= .

The roots are  y = 3,  y = 4.  (You can use the quadratic formula or the Viete's theorem).

Hence,  x = +/-  and  x = +/- 2 are the roots of the original equation.

You can put this solution on YOUR website!
I need to find the real and complex solution(s) and whether it has a multiplicity.
Problem: f(x) = x^4-7x^2+ 12
Factor::
(x^2-6)(x^2-1) = 0
x^2 = 6 or x^2 = 1
---
x = +/-sqrt(6) or x = +/-1
---------------------------
Note: I am able to solve similar equations if the exponents are sequential. However, I am unsure how to solve when they are not.
Example...x^4 - X^3 + 25x^2 - 25x = Solutions are x =1, -5i, and 5i.
-----
Rearrange::
x^4+25x^2 - (x^3+25x) = 0
x^2(x^2+25) - x(x^2+25) = 0
----
(x^2-x)(x^2+25 = 0
---
x(x-1)(x-5i)(x+5i) = 0
x = 0 or x = 1 or x = 5i or x = -5i
----------
Cheers,
Stan H.
------------

Question 992121: How do I simplify this expression: (x-8)/(x-4)+(x-5)/(x-7)-2
using this method of simplifying: (a-4)/(a-5)-(a-5)/(a-6)= [1+ 1/(a-5)]-[1+ 1/(a-6)= 1/(a-5)-1/(a-6)= -1/[(a-5)(a-6)]
Thank you

Found 2 solutions by MathTherapy, josgarithmetic:
You can put this solution on YOUR website!

How do I simplify this expression: (x-8)/(x-4)+(x-5)/(x-7)-2
using this method of simplifying: (a-4)/(a-5)-(a-5)/(a-6)= [1+ 1/(a-5)]-[1+ 1/(a-6)= 1/(a-5)-1/(a-6)= -1/[(a-5)(a-6)]
Thank you
Simplifies to: , or



You can put this solution on YOUR website!

Simplest common denominator is (x-4)(x-7) so that is the kind of terms you want.

Please show detailed workings thank you.

Found 2 solutions by ikleyn, josgarithmetic:
You can put this solution on YOUR website!
.
+ = + = .

- = - = .

Therefore,

= = = .

You can put this solution on YOUR website!
Lowest or simplest common denominator is , so multiply the complex fraction by .

Question 991597: rational expression to work on and factor polynomials completely:
81b^2-16
_________
59

Question 991560: sum of the squares of two numbers is 162. the product of the two numbers is 181. Find the numbers
You can put this solution on YOUR website!
There are no points of intersection.
.
.
.
.
.
.
.
There is no solution.

Question 991423: is pi a polynomial?
You can put this solution on YOUR website!
No, pi is a constant.
A polynomial requires at least one power of a variable.

Question 991403: What is the factored form of x squared minus x minus 6

You can put this solution on YOUR website!

Find a pair of factor of 6 that differ by 1.

Question 989965: if 3x+2y=12 and xy=6 then what is 27x^3+8y^3

You can put this solution on YOUR website!

You can also show that,

So then,

Substituting,

Question 991360: Factor the following polynomial completely.
(y + 4)^2 –2(y + 4) + 1

Found 3 solutions by MathTherapy, josgarithmetic, Fombitz:
You can put this solution on YOUR website!

Factor the following polynomial completely.
(y + 4)^2 –2(y + 4) + 1


, or