Question

The above graph represents the polynomial function $p\left(x\right)$. Using the information shown in the graph factorize $p\left(x\right)$.

• A. $p\left(x\right)=(x-4)(x+3)$
• B. $p\left(x\right)=(x-4)(x+3{)}^2$
• C. $p\left(x\right)=(x+4)(x+3)$
• D. $p\left(x\right)=(x+4)(x-3{)}^2$

Answer 1

The correct option is D $p\left(x\right)=(x+4)(x-3{)}^2$

The number of zeroes of the polynomial $p\left(x\right)=$ number of times graph of $p\left(x\right)$ intersect the $x-axis$.

When polynomial $p\left(x\right)$ touches$x-axis$ at some point let say at $x_1$, it have two equal roots at that point i.e $x_1,x_1$

Given that polynomial $f\left(x\right)$ as shown in the figure intersects $x-axis$ at one point and touches $x-axis$ at another point.

$\Rightarrow$ Polynomial $f\left(x\right)$ must have $3$ real roots in which two are equal.

The equal roots are positive and another root is negative.

Now looking at option $D$

$f\left(x\right)=(x+4)(x-3{)}^2$

It has three roots i.e $-4,3,3$

Hence, option $D$ is the correct answer.