Question

check 1 is a zero of the polynomial
$p{x} = 4x {}^{2} + 3x -1$
​

Answer 1

$\red{\boxed{\rm Solution}}$

p(x) = 4x² + 3x - 1

We know that zero of polynomial means we have equate the polynomial to 0 and we have find the value of x.

Therefore,

Zero of polynomial p(x) is,

4x² + 3x - 1 = 0

Here either we can use spilting the middle term method or we can also use quadratic equation formula.

$\red{\boxed{\rm Using \: spilting \: the \: middle \: term \: method}}$

4x² + 3x - 1 = 0

Here we have find two terms whose sum is equal to 3 and their product should be equal to the product of 4x² and - 1.

The two terms are 4x and -x

Checking whether the terms which chosen are correct or not :-

Sum of the terms should be equal to 3x

Sum of 4x and -x = 4x - x = 3x

Product of the terms should be equal to -4x²

Product of 4x and -x = -4x²

Therefore,

The terms chosen are correct

4x² + 4x - x - 1 = 0

Taking 4x common between 4x² and 4x and taking -1 common between - x and - 1.

4x(x + 1) - 1(x + 1) = 0

(4x - 1)(x + 1) = 0

If two terms (or) numbers products is equal to zero then either one of them is equal to zero or both are equal to zero.

Here we can decide which is equal to zero and which is non zero.

Therefore,

We can equating both to zero

x + 1 = 0 $\implies$ x = - 1

4x - 1 = 0 $\implies$ x = 1/4

As you see the zero of the polynomial are -1, 1/4

Hence 1 is not the zero of the polynomial p(x) = 4x² + 3x - 1

$\red{\boxed{\rm Using \: quadratic \: equation \: formula}}$

We know if ax² + bx + c = 0,

Then,

$\rm x = \frac{-b ± \sqrt{b² - 4ac} }{2a}$

Here a is the coefficient of x², b is the coefficient of x and c is constant (number).

In the polynomial p(x) = 4x² + 3x - 1

Zero of polynomial p(x) is,

4x² + 3x - 1 = 0

Here a = 4, b = 3, c = -1

Subtisute a, b, c values in above formula

$\rm \: x \: = \frac{ - 3 \pm \:\sqrt{ {( - 3)}^{2} - 4 \times 4 \times ( - 1)} }{2 \times 4}$

$\rm x = \frac{-3 ± \sqrt{9 + 16} }{8}$

$\rm x = \frac{-3± \sqrt{25} }{8}$

As there is a plus or minus operator,

That means there are two values of x

$\rm \: x \: = - \frac{ - 3 + 5}{8} = \frac{2}{8} = \frac{1}{4}$

$\rm \: x \: = \frac{ - 3 - 5}{8} = \frac{ - 8}{8} = - 1$

As you see the zero of the polynomial are -1, 1/4

Hence 1 is not the zero of the polynomial p(x) = 4x² + 3x - 1