Question
Factorize: $x^4 + x^2 + 25.$
✓
Answer
The given expression to be factorized is $x^4+x^2+25$
This can be written in the form $x^4+x^2+25=\left(x^2\right)^2+2 \cdot x^2 .5+(5)^2-9 x^2$
$=\left\{\left(x^2\right)^2+2 \cdot x^2 \cdot 5+(5)^2\right\}-(3 x)^2=\left(x^2+5\right)^2-(3 x)^2=\left(x^2+5+3 x\right)\left(x^2+5-3 x\right)$
We cannot further factorize the expression.
So, the required factorization is $x^4+x^2+25=\left(x^2+5+3 x\right)\left(x^2+5-3 x\right)$
This can be written in the form $x^4+x^2+25=\left(x^2\right)^2+2 \cdot x^2 .5+(5)^2-9 x^2$
$=\left\{\left(x^2\right)^2+2 \cdot x^2 \cdot 5+(5)^2\right\}-(3 x)^2=\left(x^2+5\right)^2-(3 x)^2=\left(x^2+5+3 x\right)\left(x^2+5-3 x\right)$
We cannot further factorize the expression.
So, the required factorization is $x^4+x^2+25=\left(x^2+5+3 x\right)\left(x^2+5-3 x\right)$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In the adjoining figure, $D$ and $E$ are respectively the midpoints of sides $AB$ and $AC$ of $\triangle\text{ABC}.$ If $PQ || BC$ and $CDP$ and $BEQ$ are straight lines then prove that$\text{ar}(\triangle\text{ABQ})=\text{ar}(\triangle\text{ACP}).$ 
→The heights (in cm) of $9$ students of a class are $148, 144, 152, 155, 160, 147, 150, 149, 145.$
Find the median height.
→Find the median height.
Rationalise the denominator of the following: $\frac{2+\sqrt{3}}{2-\sqrt{3}}$
→$ABCD$ is a trapezium in which $AB || DC, BD$ is a diagonal and $E$ is the mid-point of $AD$. A line is drawn through $E$ parallel to $AB$ intersecting $BC$ at $F$. Show that F is the mid-point of $BC.$
→Factorize: $(a - b + c)^2 + (b - c + a)^2 + 2(a - b + c)(b - c + a)$
→Assuming that $x, y, z$ are positive real numbers, simplify the following: $(\sqrt{\text{x}})^{-\frac{2}{3}}\sqrt{\text{y}^4}\div\sqrt{\text{xy}^{-\frac{1}{2}}}$
→It being given that $\sqrt{3}=1.732,\sqrt{5}=2.236,\sqrt{6}=2.449$ and $\sqrt{10}=3.162,$ find to three places of decimal, the value of the following: $\frac{1+2\sqrt{3}}{2-\sqrt{3}}$
→Express the following decimals in the form $\frac{\text{p}}{\text{q}}:$ $125.\bar{3}$
→Factorise:
$3x^3 - x^2 - 3x + 1$
→$3x^3 - x^2 - 3x + 1$
A school provides milk to the students daily in a cylindrical glasses of diameter $7\ cm$. If the glass is filled with milk upto an height of $12\ cm$, find how many litres of milk is needed to serve $1600$ students.
→