What is the value of $(a+c)(b+c)(a-d)(b-d)$ given specific roots?

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In summary, "Evaluate (a+c)(b+c)(a-d)(b-d)" is a mathematical equation that involves finding the value of the product of four terms. The equation can be simplified using the distributive property and combining like terms. There is no specific order in which the terms need to be evaluated, and any variables can be used as long as the correct steps are followed.
  • #1
anemone
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Given that $a$ and $b$ are the roots of the equation $x^2+2000x+1=0$ whereas $c$ and $d$ are the roots of the equation $x^2-2008x+1=0$.

Evaluate $(a+c)(b+c)(a-d)(b-d)$.
 
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  • #2
My attempt

We are given a,b are roots of equation so $f(x) = (x -a)(x-b) = x^2 + 2000x +1 = 0$
so $(a+c)(b+c) = (-c - a)(-c-b) = f(-c) = c^2 - 2000c + 1$

as c is root of $x^2-2008x + 1 = 0$ so $c^2 - 2000c +1 = 0$

so $(a+c)(b+c) = 0$ and multiplying by $(a-d)(b-d)$ we get $(a+c)(b+c)(a-d)(b-d) = 0$
 
  • #3
Hi Kali...

Your answer is not correct...I am sorry.
 
  • #4
Thanks anemone., I mixed up 2000 and 2008. I have solved it incorrectly
 
  • #5
anemone said:
Given that $a$ and $b$ are the roots of the equation $x^2+2000x+1=0$ whereas $c$ and $d$ are the roots of the equation $x^2-2008x+1=0$.

Evaluate $(a+c)(b+c)(a-d)(b-d)$.

Let's first look at the product:

\(\displaystyle (a+c)(b-d)=ab-ad+bc-cd\)

By Viete, we know:

\(\displaystyle ab=cd=1\)

Hence:

\(\displaystyle (a+c)(b-d)=bc-ad\)

Next, let's look at the product:

\(\displaystyle (b+c)(a-d)=ab-bd+ac-cd=ac-bd\)

And so we find:

\(\displaystyle (a+c)(b+c)(a-d)(b-d)=(bc-ad)(ac-bd)=abc^2-b^2cd-a^2cd+abd^2=c^2+d^2-(a^2+b^2)\)

Now, also by Viete, we have:

\(\displaystyle a+b=-2000\)

Hence:

\(\displaystyle a^2+2ab+b^2=2000^2\implies a^2+b^2=2000^2-2\)

Likewise:

\(\displaystyle c+d=2008\)

\(\displaystyle c^2+2cd+d^2=2008^2\implies c^2+d^2=2008^2-2\)

And so:

\(\displaystyle (a+c)(b+c)(a-d)(b-d)=(2008^2-2)-(2000^2-2)=2008^2-2000^2=8(4008)=32064\)
 
  • #6
Very good job, MarkFL!
 
  • #7
By Vieta's formula, we have $a+b=-2000; ab=1; cd=1$.

$\begin{align*}(a+c)(b+c)(a-d)(b-d)&=[ab+(a+b)c+c^2][ab-(a+b)d+d^2]\\&=(1-2000c+c^2)(1+2000d+d^2)\\&=(c^2-2008c+1+8c)(d^2-2008d+1+4008d)\\&=8c(4008d)\\&=32064\end{align*}$
 

Related to What is the value of $(a+c)(b+c)(a-d)(b-d)$ given specific roots?

1. What is the purpose of evaluating (a+c)(b+c)(a-d)(b-d)?

The purpose of evaluating (a+c)(b+c)(a-d)(b-d) is to simplify and find the value of the expression. This can be useful in solving equations, graphing functions, and making predictions in various scientific and mathematical fields.

2. How do you evaluate (a+c)(b+c)(a-d)(b-d)?

To evaluate (a+c)(b+c)(a-d)(b-d), you can use the distributive property and the rules of algebra to expand the expression and combine like terms. This will result in a simplified expression that can be further evaluated depending on the given values of a, b, c, and d.

3. Can (a+c)(b+c)(a-d)(b-d) be evaluated without knowing the values of a, b, c, and d?

No, (a+c)(b+c)(a-d)(b-d) cannot be evaluated without knowing the values of a, b, c, and d. These variables represent unknown quantities and must be assigned specific values in order to evaluate the expression.

4. What is the order of operations when evaluating (a+c)(b+c)(a-d)(b-d)?

The order of operations when evaluating (a+c)(b+c)(a-d)(b-d) is the same as in any other mathematical expression. First, you must evaluate any parentheses or brackets, then perform any multiplications or divisions from left to right, and finally, perform any additions or subtractions from left to right.

5. Can (a+c)(b+c)(a-d)(b-d) be rewritten in a simpler form?

Yes, (a+c)(b+c)(a-d)(b-d) can be rewritten in a simpler form by expanding and combining like terms. This can result in a polynomial expression with fewer terms or a completely simplified expression with a numerical value.

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