Question

Question 1(i)
Find the zeroes of the following polynomials by factorization method and verify the relations between the zeroes and the coefficient of the polynomials
(i) $4x^2–3x–1$

Answer 1

Let f(x) = $4x^2–3x–1$
$=4x^2–4x+x–1$ [by splitting the middle term]
$=4x(x–1)+1(x–1)$
$=(x–1)(4x+1)$
So, the value of $4x^2–3x–1$ is zero when x – 1 = 0 or 4x + 1 = 0 i.e., when x = 1 or $x=-\frac{1}{4}$
So, the zeroes of $4x^2–3x–1$ are $1$ and $-\frac{1}{4}$

Now to verify the relations between the zeroes and the coefficient:
Sum of zeroes = $\left(\frac{-b}{a}\right)=1-\frac{1}{4}=\frac{3}{4}=\frac{-(-3)}{4}$
$=(-1)\frac{Coefficientofx}{Coefficientof{x}^2}$.

And product of zeroes = $\left(\frac{c}{a}\right)=\left(1\right)(-\frac{1}{4})=(-\frac{1}{4})$
$=(-1{)}^2\frac{constantterm}{Coefficientof{x}^2}$
Hence, verified.