Question

Find the factors of the polynomials given below.
(i) 2x² + x – 1

(ii) 2m² + 5m – 3

(iii) 12x² + 61x + 77

(iv) 3y² – 2y – 1

(v) $\sqrt{3}{x}^2+4x+\sqrt{3}$

(vi) $\frac{1}{2}{x}^2-3x+4$

Answer 1

(i)
$$\begin{gathered} 2x^2+x-1 \\ =2x^2+2x-x-1 \\ =2x\left(x+1\right)-1\left(x+1\right) \\ =\left(x+1\right)\left(2x-1\right) \end{gathered}$$
(ii)
$$\begin{gathered} 2m^2+5m-3 \\ =2m^2+6m-m-3 \\ =2m\left(m+3\right)-1\left(m+3\right) \\ =\left(m+3\right)\left(2m-1\right) \end{gathered}$$
(iii)
$$\begin{gathered} 12x^2+61x+77 \\ =12x^2+28x+33x+77 \\ =4x\left(3x+7\right)+11\left(3x+7\right) \\ =\left(3x+7\right)\left(4x+11\right) \end{gathered}$$
(iv)
$$\begin{gathered} 3y^2-2y-1 \\ =3y^2-3y+y-1 \\ =3y\left(y-1\right)+1\left(y-1\right) \\ =\left(y-1\right)\left(3y+1\right) \end{gathered}$$
(v)
$$\begin{gathered} \sqrt{3}{x}^2+4x+\sqrt{3} \\ =\sqrt{3}{x}^2+3x+x+\sqrt{3} \\ =\sqrt{3}x\left(x+\sqrt{3}\right)+1\left(x+\sqrt{3}\right) \\ =\left(x+\sqrt{3}\right)\left(\sqrt{3}x+1\right) \end{gathered}$$
(vi)
$$\begin{gathered} \frac{1}{2}{x}^2-3x+4 \\ =\frac{1}{2}{x}^2-2x-x+4 \\ =\frac{1}{2}x\left(x-4\right)-1\left(x-4\right) \\ =\left(x-4\right)\left(\frac{1}{2}x-1\right) \end{gathered}$$