Question

ind the quadratic polynomial whose zeroes are 2 and -6 verify the relation between the coefficients and the zeroes of the polynomial

Answer 1

AnsWer:-

↝α+β=2+(-6)

↝α+β=-4

↝αβ=2×-6

↝αβ=-12

✪Using the Formula

→k[x²-(α+β)x+αβ]

↝k[x²-(-4)x+(-12)]

↝k[x²+4x-12]

•Let k=1•

↝1[x²+4x-12]

☞x²+4x-12 is the Polynomial.

*Since The Zeros Form a Polynomial,It Verifies the Relation b/w coefficients and the zeros of the polynomial.*

$\rule{200}{2}$

Answer 2

Answer:

• $\alpha$ = 2
• $\beta$ = - 6

$\underline{\bigstar\:\textbf{Standard Form of Quadratic Polynomial :}}$

$\dashrightarrow\rm\:\:x^2-(Sum\:of\: Zeroes)x+(Product\:of\:Zeroes)\\\\\\\dashrightarrow\rm\:\:x^2-(\alpha+\beta)x + (\alpha\beta)\\\\\\\dashrightarrow\rm\:\:x^2-[2+ (-6)]x+[2\times-6]\\\\\\\dashrightarrow\rm\:\:x^2-(-4)x+(-12)\\\\\\\dashrightarrow\rm\:\:x^2+4x-12\\\\{\qquad\textsf{Here, a = 1;\quad b = 4;\quad c = - 12}}$

$\rule{150}{1}$

$\underline{\bigstar\:\textsf{Relation b/w zeroes and coefficient :}}$

$\qquad\underline{\bf{\dag}\:\:\textsf{Sum of Zeroes :}}\\\dashrightarrow\tt\:\: \alpha+\beta = \dfrac{-\:b}{a}\\\\\\\dashrightarrow\tt\:\: 2 +(-6) = \dfrac{-4}{1}\\\\\\\dashrightarrow\:\:\underline{\boxed{\red{\tt -\:4=-\:4}}}\\\\\\{\qquad\underline{\bf{\dag}\:\:\textsf{Product of Zeroes :}}}\\\\\dashrightarrow\tt\:\: \alpha \times \beta = \dfrac{c}{a}\\\\\\\dashrightarrow\tt\:\: 2\times (-6) = \dfrac{-12}{1}\\\\\\\dashrightarrow\:\:\underline{\boxed{ \red{\tt -\:12=-\:12}}}$