Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebra-precalculus]

For questions about algebra and precalculus topics, which include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomial, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

This tag is for questions typically taught in precalculus, as well as elementary algebra.

These topics include linear, exponential, logarithmic, polynomial, rational, and trigonometric functions; conic sections, binomials, surds, graphs and transformations of graphs, solving equations and systems of equations; and other symbolic manipulation topics.

47234 questions
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Sum of the $11^\mathrm{th}$ power of the roots of the equation $x^5+5x+1=0$

Find the sum of the $11^\mathrm{th}$ power of the all roots of the equation $$ x^5+5x+1=0 $$ My Attempt: Let $R=\{\alpha,\beta,\gamma,\delta,\mu\}$ be the set of all roots of the equation ${x^5+5x+1=0}$, and let $x\in R$. Then we…
juantheron
  • 53,015
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Sine and cosine expression

Find the greatest possible value of $5\cos x + 6\sin x$. I attempted to solve this using graphing, however, the answer appears to be an ugly irrational. Is there a better method of solving this problem? Thank you.
math-sd
  • 691
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Formula for completing the square?

My math teacher said that this was the formula for completing the square. Original function: $$ax^2 + bx + c$$ Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$ However, using this formula I'm not getting the same answers…
Yep
  • 229
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Show that $(x^2-yz)^3+(y^2-zx)^3+(z^2-xy)^3-3(x^2-yz)(y^2-zx)(z^2-xy)$ is a perfect square and find its square root.

Show that $(x^2-yz)^3+(y^2-zx)^3+(z^2-xy)^3-3(x^2-yz)(y^2-zx)(z^2-xy)$ is a perfect square and find its square root. My work: Let, $x^2-yz=a,y^2-zx=b,z^2-xy=c$. So, we can…
Hawk
  • 6,540
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Trying to solve an odd equation minimum.

Here is a math puzzle I've been working on - The set-up - I am a dog, 50 feet from the water. A rubber duck is in the water 50 feet from shore and 140 feet to my right. This makes a triangle base of 140 ft, height 100 ft. I can run 30fps on land,…
JTP - Apologise to Monica
  • 3,102
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How to know what is the degree of the remainder

The example question is Find the remainder when $8x^4+3x-1$ is divided by $2x^2+1$ The answer did something like $$8x^4+3x-1=(2x^2+1)(Ax^2+Bx+C)+(Dx+E)$$ Where $(Ax^2+Bx+C)$ is the Quotient and $(Dx+E)$ the remainder. I believe the degree of…
Jiew Meng
  • 4,593
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How prove $a*b=b*a$

let $x,y$ be any real numbers,define $*$,such $$x=(x*y)*y=y*(y*x)$$ fo any $a,b$,show that $$a*b=b*a$$ My try: $$x=(x*y)*y=y*(y*x)$$ then $$y=(y*x)*x=x*(x*y)$$ so $$y*x=x*(x*y)*x$$ and $$x*y=y*(y*x)*y$$ then I can't,Thank you
math110
  • 93,304
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Factoring a Cubic Polynomial

I've been trying to understand how ${x^3-12x+9}$ factors to $(x-3) (x^2+3 x-3)$ What factoring rule does this follow? The net result seems to be similar to what is attained through the sum/difference of cubes factoring pattern, but the signs are…
michaelmichael
  • 173
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Finding a,b,c,d in a quartic expression

Let $p(x)=x^4+ax^3+bx^2+cx+d$ where a,b,c,d are constants. If $p(1)=10$, $p(2)=20$, $p(3)=30$, compute $\frac {p(12)+p(-8)}{10}$. I have tried so far. \begin{align} a+b+c+d=&9\\8a+4b+2c+d=&4\\27a+9b+3c+d=&-51 \end{align} Manipulating these, I got…
Tejas
  • 2,082
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Real solutions of the equation $x = \sqrt{3-x} \cdot \sqrt{4-x} + \sqrt{4-x} \cdot \sqrt{5-x} + \sqrt{5-x} \cdot \sqrt{3-x}$

Solve the equation $$x = \sqrt{3-x} \cdot \sqrt{4-x} + \sqrt{4-x} \cdot \sqrt{5-x} + \sqrt{5-x} \cdot \sqrt{3-x},$$ where $x \in \mathbb{R}$.
Hung Nguyen
  • 1,841
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Show that if $x^2+x+1=0$, then $x^{26}+\dfrac{1}{x^{26}}=-1$

I am trying to answer the following precalculus question: Show that if $x^2+x+1=0$, then $A=x^{26}+\dfrac{1}{x^{26}}=-1$. Let's multiply the first equality (that we know is true) by $(x-1)$. We can do that as $x$ is obviously $\ne1$, $x=1$ isn't a…
yinivem462
  • 644
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Which is greatest in the sequence $1, \sqrt 2 , \sqrt[3] 3 , \sqrt[4] 4 \cdot \cdot \cdot?$

This problem is from Ivan Niven's "Maxima and Minima Without Calculus". What is another way to find this? The solution from the book was: Note that $\large \sqrt[4] 4 =\sqrt 2$ so this hints that $\sqrt[3] 3$ is the largest. Next the book proved…
Ovi
  • 23,737
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Solve the equation for x, y and z: $\sqrt{x-y+z}=\sqrt x - \sqrt y + \sqrt z$

I am having some trouble with this problem, Solve for $x,y,$ and $z$. $$\sqrt{x-y+z}=\sqrt x - \sqrt y + \sqrt z$$ Here is my work so far, $$x - y +z = x+y+z-2\sqrt{xy} + 2\sqrt{xz}- 2\sqrt{zy}$$ $$2y-2\sqrt{xy} + 2\sqrt{xz}- 2\sqrt{zy} = 0…
Saiichi
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Why is the derivative of sine the cosine in radians but not in degrees?

In radians the derivative of sine is the cosine. But why isn't this the same in degrees? According to this I'd first have to convert it to radians before this works. But when you look at the graph of sine whether the x is in degrees or radians, the…
user1534664
  • 1,272
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How to understand proof of a limit of a function?

Given the following function: $$ f(x)=\left\{ \begin{array} {cc} 0, & x \text{ irrational, } 0
mauna
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