Question

Find the zeroes of the polynomial (3x)2+4x+1 and verify the relation between zeroes and the cofficient

Answer 1

Step-by-step explanation:

Find the zeros,

$\longrightarrow$ 3x² + 4x + 1

$\longrightarrow$ 3x² + 3x + x + 1

$\longrightarrow$ 3x(x + 1) + 1(x + 1)

$\longrightarrow$ (3x + 1)(x + 1)

Zeros =

$\longrightarrow$ 3x + 1 = 0

$\longrightarrow$ 3x = –1

$\longrightarrow$ x = –1/3

First zero = –1/3

$\longrightarrow$ x + 1 = 0

$\longrightarrow$ x = –1

Second zero = –1

__________________

Verifying the relationship between zeros and coefficients :

Let the zeros of the polynomial be α and β.

In the polynomial,

• a = 3
• b = 4
• c = 1

Sum of zeros =

$\longrightarrow$ –1 + –1/3

$\longrightarrow$ –4/3

α + β =

$\longrightarrow$ –b/a

$\longrightarrow$ –4/3

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Product of zeros =

$\longrightarrow$ –1 × –1/3

$\longrightarrow$ 1/3

α × β =

$\longrightarrow$ c/a

$\longrightarrow$ 1/3

Hence verified.

Answer 2

Information given to us:

• Polynomial (3x) 2+4x + 1

Need to be calculated:

• We have to find out the zeroes of the given above polynomial.
• Also we have to verify the relation between zeroes and the cofficient

Calculations we are performing:

1st part of question is to find the zeroes of polynomial:-

In equation f(x) = 3x² + 4x + 1 , is said to be a function of variable x as the value of f(x) depends on the value of x.

Finding out the zeroes:-

→ 3x² + 4x + 1

Splitting middle term:

→ 3x² + 3x + x + 1

Grouping terms:

→ 3x (x + 1) + 1 (x + 1)

We gets,

→ (3x + 1) (x + 1)

Now, calculating the first zero :-

Equation given,

→ 3x + 1

The given equation would be equal to zero,

→ 3x + 1 = 0

Transposing sides from L.H.S. to R.H.S:

→ 3x = -1

By dividing we gets,

→ x = -1/3

Thus, we came to knew about the first zero of the polynomial that is -1/3.

Now, calculating the second zero :-

Equation given,

→ x + 1

The given equation would be equal to zero,

→ x + 1 = 0

Transposing sides from L.H.S. to R.H.S:

→ x = -1

Thus, we came to knew about the second zero of the polynomial that is -1.

2nd part of the question is to verify the relation between zeroes and the coefficient:-

In this we know about the values of a, b, and c

Which are,

• a is 3
• b is 4
• c is 1

Solving product of zeroes:

→ -1 × 4/3

→ 1/3

Solving sum of zeroes:

→ -1 + (-1/3)

→ (-1/1) + (-1/3)

→ (-3 - 1 ) / 3

We know that,

• Two minus gives out plus

→ -4/3

Therefore, zeroes of polynomials are -1 and -1/3 and the relation between zeroes and the cofficient is verified.