Question

Find the zeros, the sum and the product of zeros of $p\left(x\right)=4x^2+12x+5$.

Answer 1

In the given polynomial $4x^2+12x+5$,

The first term is $4x^2$ and its coefficient is $4$.

The middle term is $12x$ and its coefficient is $12$.

The last term is a constant term $5$.

Multiply the coefficient of the first term by the constant $4\times 5=20$.

We now find the factor of $20$ whose sum equals the coefficient of the middle term, which is $12$ and then factorize the polynomial $4x^2+12x+5$and equate it to $0$ as shown below:

$4x^2+12x+5=0\Rightarrow 4x^2+2x+10x+5=0\Rightarrow 2x(2x+1)+5(2x+1)=0\Rightarrow (2x+1)(2x+5)=0\Rightarrow 2x=-1,2x=-5\Rightarrow x=-\frac{1}{2},x=-\frac{5}{2}$

Therefore, the zeroes of the polynomial $4x^2+12x+5$ are $-1,\frac{4}{3}$.

Now, the sum and the product of the zeroes of the given quadratic polynomial is as follows:

$-\frac{1}{2}+(-\frac{5}{2})=-\frac{1}{2}-\frac{5}{2}=-\frac{6}{2}=-3$

$(-\frac{1}{2})\times (-\frac{5}{2})=\frac{1}{2}\times \frac{5}{2}=\frac{5}{4}$

Hence, the sum of the zeroes is $-3$ and the product of the zeroes is $\frac{5}{4}$.