Question

Given that $P\left(x\right)andQ\left(x\right)$ are polynomials of degree $3$ with real coefficients, which one of the following is not true ?

• A. $deg\left[P\right(x)\times Q(x\left)\right]=6$
• B. $deg\left[P\right(x)+Q(x\left)\right]\le 3$
• C. $deg\left[p\right(x)-Q(x\left)\right]\le 3$
• D. $deg\left[\begin{array}{c}\left[P\right(x\left){]}^2\right[Q\left(x\right){]}^2\end{array}\right]=15$

Answer 1

The correct option is D $deg\left[\begin{array}{c}\left[P\right(x\left){]}^2\right[Q\left(x\right){]}^2\end{array}\right]=15$

$deg\left(\begin{array}{c}p\left(x\right)\times q\left(x\right)\end{array}\right)=6$ possible
$deg\left(\begin{array}{c}p\left(x\right)+q\left(x\right)\end{array}\right)=3$ possible
$deg\left(\begin{array}{c}p\left(x\right)-q\left(x\right)\end{array}\right)=3$ possible
$deg{\left(\begin{array}{c}p\left(x\right)\end{array}\right)}^2.q{\left(\begin{array}{c}\left(x\right)\end{array}\right)}^2=12$ but $15$ is given $4^{th}$ option is not possible