Question

Find the product of the following polynomials and state the degree of their product.

(i) x² + 3x + 1; 2x $-$ 3
(ii) 3x² + 5x ; x² + 2x + 1
(iii) x³ + 4x + 2 ; x² + x + 5
(iv) x³ $-$ 1 ; x² $-$ x + 4
(v) 2y² + 3 ; 3y³ + 1

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{We}\mathrm{have}\left(x^2+3x+1\right)\left(2x-3\right) \\ =x^2\left(2x-3\right)+3x\left(2x-3\right)+1\left(2x-3\right) \\ =2x^3-3x^2+6x^2-9x+2x-3 \\ =2x^3+3x^2-7x-3 \\ \therefore \mathrm{Degree}=3 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{We}\mathrm{have}\left(3x^2+5x\right)\left(x^2+2x+1\right) \\ =3x^2\left(x^2+2x+1\right)+5x\left(x^2+2x+1\right) \\ =3x^4+6x^3+3x^2+5x^3+10x^2+5x \\ =3x^4+11x^3+13x^2+5x \\ \therefore \mathrm{Degree}=4 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{We}\mathrm{have}\left(x^3+4x+2\right)\left(x^2+x+5\right) \\ =x^3\left(x^2+x+5\right)+4x\left(x^2+x+5\right)+2\left(x^2+x+5\right) \\ =x^5+x^4+5x^3+4x^3+4x^2+20x+2x^2+2x+10 \\ =x^5+x^4+9x^3+6x^2+22x+10 \\ \therefore \mathrm{Degree}=5 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{We}\mathrm{have}\left(x^3-1\right)\left(x^2-x+4\right) \\ =x^3\left(x^2-x+4\right)-1\left(x^2-x+4\right) \\ =x^5-x^4+4x^3-x^2+x-4 \\ \therefore \mathrm{Degree}=5 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)\mathrm{We}\mathrm{have}\left(2y^2+3\right)\left(3y^3+1\right) \\ =2y^2\left(3y^3+1\right)+3\left(3y^3+1\right) \\ =6y^5+2y^2+9y^3+3 \\ =6y^5+9y^3+2y^2+3 \\ \therefore \mathrm{Degree}=5 \end{gathered}$$