Write the expression as a polynomial (x+1)(x+2)(y–3)

1 answer:

6 0

$\text{Use FOIL method:}\ (a+b)(c+d)=ac+ad+bc+bd\\\\(x+1)(x+2)(y-3)=[(x)(x)+(x)(2)+(1)(x)+(1)(2)](y-3)\\\\=(x^2+2x+x+2)(y-3)=(x^2+3x+2)(y-3)\\\\=(x^2)(y)+(x^2)(-3)+(3x)(y)+(3x)(-3)+(2)(y)+(2)(-3)\\\\=\boxed{x^2y-3x^2+3xy-9x+2y-6}$

You might be interested in

Can you help me with these problems please

Andreyy89

1.534
2.168
3.230
4.408
5.78
6.642
7.3
81016
9.1900
10.8 times (40 plus 5)

7 0

2 years ago

Sadie computes the perimeter of a rectangle by adding the length, l , and width, w , and doubling this sum. Eric computes the pe

larisa [96]

a) P = 2(l+w) Sadie

P =2l+2w Eric

b) Sadie : P = 2(30+75) = 2(105) = 210

Eric: P = 2(30) +2(75) = 60+150 =210

4 0

2 years ago

Use lagrange multipliers to find the shortest distance, d, from the point (4, 0, −5 to the plane x y z = 1

Varvara68 [4.7K]

I assume there are some plus signs that aren't rendering for some reason, so that the plane should be $x+y+z=1$ .

You're minimizing $d(x,y,z)=\sqrt{(x-4)^2+y^2+(z+5)^2}$ subject to the constraint $f(x,y,z)=x+y+z=1$ . Note that $d(x,y,z)$ and $d(x,y,z)^2$ attain their extrema at the same values of $x,y,z$ , so we'll be working with the squared distance to avoid working out some slightly more complicated partial derivatives later.

The Lagrangian is

$L(x,y,z,\lambda)=(x-4)^2+y^2+(z+5)^2+\lambda(x+y+z-1)$

Take your partial derivatives and set them equal to 0:

$\begin{cases}\dfrac{\partial L}{\partial x}=2(x-4)+\lambda=0\\\\\dfrac{\partial L}{\partial y}=2y+\lambda=0\\\\\dfrac{\partial L}{\partial z}=2(z+5)+\lambda=0\\\\\dfrac{\partial L}{\partial\lambda}=x+y+z-1=0\end{cases}\implies\begin{cases}2x+\lambda=8\\2y+\lambda=0\\2z+\lambda=-10\\x+y+z=1\end{cases}$

Adding the first three equations together yields

$2x+2y+2z+3\lambda=2(x+y+z)+3\lambda=2+3\lambda=-2\implies \lambda=-\dfrac43$

and plugging this into the first three equations, you find a critical point at $(x,y,z)=\left(\dfrac{14}3,\dfrac23,-\dfrac{13}3\right)$ .

The squared distance is then $d\left(\dfrac{14}3,\dfrac23,-\dfrac{13}3\right)^2=\dfrac43$ , which means the shortest distance must be $\sqrt{\dfrac43}=\dfrac2{\sqrt3}$ .