Question

2x^+5x-12
find the zeros of the polynomial and verify the relatn blw the zeros and coefficient ​

Answer 1

Hey , mate

Your answer is here

Answer 2

Correct QuestioN :

2x² + 5x - 12 is polynomial, find the zeros of the polynomial and verify the relation between the zeros and coefficient

SolutioN :

Let, Zeros be.

$\tt \bullet \: \: \: \: \: \alpha \: \: and \: \: \beta$

We have, Expression.

$\tt \dagger \: \: \: \: \: 2 {x}^{2} + 5x - 12.$

# We know, that.

$\tt \blacksquare \: Sum \: of \: Zeros. \\ \tt \alpha + \beta = \frac{ - b}{a} = \dfrac{coefficient \: of \: x}{coefficient \: of \: {x}^{2} }$

$\tt \blacksquare \: Product \: of \: Zeros. \\ \tt \alpha \beta = \frac{ c}{a} = \dfrac{constant \: term}{coefficient \: of \: {x}^{2} }$

Now,

★ Sum of Zeros

→ -b /a

→ -5 /2.

★ Product of Zeros.

→ c / a

→ - 12 /2.

→ - 6

Let's try to find there both zeros ( Splitting the middle term )

→ 2x² + 5x - 12.

→ 2x² + 8x - 3x - 12.

→ 2x( x + 4 ) - 3( x + 4 )

→ ( 2x - 3 )( x + 4 )

★ Either,

→ 2x - 3 = 0.

→ 2x = 3.

→ x = 3/2.

★ Or,

→ x + 4 = 0.

→ x = - 4.

Now,

Let,

• alpha = 3/2
• beta = - 4.

Let's Verify ( Zeros and coefficient )

$\tt Sum \: of \: Zeros = \alpha + \beta$

$\tt : \implies \dfrac{ - 5}{2} = \dfrac{3}{2} + - 4$

$\tt : \implies \dfrac{ - 5}{2} = \dfrac{3 - 8}{2}$

$\tt : \implies \dfrac{ - 5}{2} = \dfrac{ - 5}{2}$

$\tt Product \: of \: Zeros = \alpha \beta$

$\tt : \implies - 6= \dfrac{3}{2} \times - 4$

$\tt : \implies - 6 = - 6.$

Hence Verify.