Is it correct that the sum of the factors of the polynomial $4b^2c^2-(b^2+c^2-a^2)^2$ is equal to $2(a+b+c)$?
Given: The polynomial: $4b^2c^2-(b^2+c^2-a^2)^2$.
To do: To check whether the sum of the factors of the polynomial $4b^2c^2-(b^2+c^2-a^2)^2$ is equal to $2(a+b+c)$.
Solution:
$4b^2c^2-(b^2+c^2-a^2)^2$
$=( 2bc)^2-( b^2+c^2-a^2)^2$
$=( 2bc+b^2+c^2-a^2)(2bc-b^2-c^2+a^2))$
$=( ( b+c)^2-a^2)( a^2-( b-c)^2)$
$=( b+c-a)(b+c+a)(a-b+c)(a+b-c)$
Therefore, $( b+c-a),\ (b+c+a),\ (a-b+c),\ (a+b-c)$ are the factors of $4b^2c^2-(b^2+c^2-a^2)^2$.
The sum of the factors$=( b+c-a)+(b+c+a)+(a-b+c)+(a+b-c)$
$=b+c-a+b+c+a+a-b+c+a+b-c$
$=2a+2b+2c$
$=2( a+b+c)$
Thus, it is correct that the sum of the factors of the polynomial $4b^2c^2-(b^2+c^2-a^2)^2$ is equal to $2(a+b+c)$.
Related Articles
- Factorize $(a^2-b^2-c^2) ^2-4b^2c^2$.
- If the roots of the equation $(a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$.
- If \( a+b=5 \) and \( a b=2 \), find the value of(a) \( (a+b)^{2} \)(b) \( a^{2}+b^{2} \)(c) \( (a-b)^{2} \)
- Given that $sin\ \theta = \frac{a}{b}$, then $cos\ \theta$ is equal to(A) \( \frac{b}{\sqrt{b^{2}-a^{2}}} \)(B) \( \frac{b}{a} \)(C) \( \frac{\sqrt{b^{2}-a^{2}}}{b} \)(D) \( \frac{a}{\sqrt{b^{2}-a^{2}}} \)
- Simplify:$(a + b + c)^2 + (a - b + c)^2 + (a + b - c)^2$
- Which of the following is the correct electronic configuration of sodium?(a) 2, 8, 1 (b) 8, 2, 1 (c) 2, 1, 8 (d) 2, 8, 2
- Find the greatest common factor of the terms in the expression $3a^2b^2+4b^2c^2+12a^2b^2c^2$.
- If $a=2$ and $b=-2$ find the value of $(i)$. $a^2+b^2$$(ii)$. $a^2+ab+b^2$$(iii)$. $a^{2}-b^2$
- Factorize:$(a – b + c)^2 + (b – c + a)^2 + 2(a – b + c) (b – c + a)$
- If $a ≠ b ≠ c$, prove that the points $(a, a^2), (b, b^2), (c, c^2)$ can never be collinear.
- Show that the sum of an AP whose first term is \( a \), the second term \( b \) and the last term \( c \), is equal to \( \frac{(a+c)(b+c-2 a)}{2(b-a)} \)
- Count number of triplets (a, b, c) such that a^2 + b^2 = c^2 and 1
- If \( x=\frac{\sqrt{a^{2}+b^{2}}+\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}-\sqrt{a^{2}-b^{2}}} \), then prove that \( b^{2} x^{2}-2 a^{2} x+b^{2}=0 \).
- The correct electronic configuration of a chloride ion is :(a) 2, 8 (b) 2, 8, 4 (c) 2, 8, 8 (d) 2, 8, 7
- The common difference of the A.P: $-4,\ -2,\ 0,\ 2, \ldots$ is$( a).\ 2\ ( b).\ -2\ ( c).\ \frac{1}{2}\ ( d).\ -\frac{1}{2}$.
Kickstart Your Career
Get certified by completing the course
Get Started