Question
Using factor theorem, factorize the following polynomials:
$y^3 - 7y + 6$
$y^3 - 7y + 6$
✓
Answer
Let $f(y)=y^3-7 y+6$
The factors of constant term in $f(y)$ are $\pm 1, \pm, 2, \pm 3$ and $\pm 6$.
We have,
$f (1)=1-7+6=0$
$\Rightarrow( y -1)$ is a factor of $f ( y )$
$f (-1)=-1+7+6=12$
$\Rightarrow(y+1)$ is a factor of $f(y)$
$f(2)=8-14+6=0$
$\Rightarrow( y -2)$ is a factor of $f ( y )$
$f(-2)=-8+14+6=12$
$\Rightarrow(y+2)$ is not a factor of $f ( y )$
$f(3)=27-21+6=12$
$\Rightarrow(y-3)$ is not a factor of $f(y)$
$f(-3)=-27+21+6=0$
$\Rightarrow(y+3)$ is a factor of $f(y)$
Since $f(y)$ is a polynomial of degree 3 . So, it cannot have more than 3 linear factors.
Thus, factors of $f(y)$ are $(y-1)(y-2)$ and $(y+3)$.
Therefore,
$f(y)=k(y-1)(y-2)(y+3)$
$y^3-7 y+6=k(y-1)(y-2)(y+3) \ldots(1)$
Putting $y=0$ on both sides, we get,
$6=k(-1)(-2)(3)$
$6=6 k$
$k=1$
Substituting $k=1$ in (1), we get,
$y^3-7 y+6=(y-1)(y-2)(y+3)$
The factors of constant term in $f(y)$ are $\pm 1, \pm, 2, \pm 3$ and $\pm 6$.
We have,
$f (1)=1-7+6=0$
$\Rightarrow( y -1)$ is a factor of $f ( y )$
$f (-1)=-1+7+6=12$
$\Rightarrow(y+1)$ is a factor of $f(y)$
$f(2)=8-14+6=0$
$\Rightarrow( y -2)$ is a factor of $f ( y )$
$f(-2)=-8+14+6=12$
$\Rightarrow(y+2)$ is not a factor of $f ( y )$
$f(3)=27-21+6=12$
$\Rightarrow(y-3)$ is not a factor of $f(y)$
$f(-3)=-27+21+6=0$
$\Rightarrow(y+3)$ is a factor of $f(y)$
Since $f(y)$ is a polynomial of degree 3 . So, it cannot have more than 3 linear factors.
Thus, factors of $f(y)$ are $(y-1)(y-2)$ and $(y+3)$.
Therefore,
$f(y)=k(y-1)(y-2)(y+3)$
$y^3-7 y+6=k(y-1)(y-2)(y+3) \ldots(1)$
Putting $y=0$ on both sides, we get,
$6=k(-1)(-2)(3)$
$6=6 k$
$k=1$
Substituting $k=1$ in (1), we get,
$y^3-7 y+6=(y-1)(y-2)(y+3)$
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
The mean of nine numbers is 77. If one more number is added to it, then the mean increases by 5. Find the number added in the data.
A hand fan is made by sticking 10 equal size triangular strips of two different types of paper as shown in the figure. The dimensions of equal strips are 25cm, 25cm and 14cm. Find the area of each type of paper needed to make the hand fan.
If the ratio of mean and median of a certain data is 2 : 3, then find the ratio of its mode and mean.
Find the perimeter and area of the quadrilateral ABCD in which AB = 17cm, AD = 9cm, CD = 12cm, $\angle\text{ACB}=90^\circ$ and AC = 15cm.
→In the given figure, AB || CD. Prove that $\angle\text{BAE}-\angle\text{ECD}=\angle\text{AEC}.$
→Each edge of a cube is increased by $50 \%$. Find the percentage increase in the surface area of the cube.
→Marks scored by 40 students of class IX in mathematics are given below:
81, 55, 68, 79, 85, 43, 29, 68, 54, 73, 47, 35, 72, 64, 95, 44, 50, 77, 64, 35, 79, 52, 45, 54, 70, 83, 62, 64, 72, 92, 84, 76, 63, 43, 54, 38, 73, 68, 52, 54.
Prepare a frequency distribution with class size of 10 marks.
81, 55, 68, 79, 85, 43, 29, 68, 54, 73, 47, 35, 72, 64, 95, 44, 50, 77, 64, 35, 79, 52, 45, 54, 70, 83, 62, 64, 72, 92, 84, 76, 63, 43, 54, 38, 73, 68, 52, 54.
Prepare a frequency distribution with class size of 10 marks.
If $\text{x}-\frac{1}{\text{x}}=3+2\sqrt2,$ find the value of $\text{x}^3-\frac{1}{\text{x}^3}.$
→Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.
Read the bar graph given in Fig. and answer the following questions:
- What information is given by the bar graph?
- Which Door darshan center covers maximum area? Also, tell the covered area.
- What is the difference between the areas covered by the centers at Delhi and Bombay?
- Which Door darshan centers are in U.P. State? What are the areas covered by them?