Question

If (x-4) is the factor of quadratic polynomial p(x) and 2 is a zero of p(x), then find the polynomial p(x). ​

Answer 1

Solution:

Given That: p(x) is a quadratic polynomial.

This implies that p(x) is in the form of ax² + bx + c

Now, (x - 4) is a factor of p(x). Therefore, by factor theorem –

→ p(4) = 0

→ 4 is a zero of p(x)

→ Also, 2 is a zero.

Now, 4 and 2 are the zeros of p(x). Therefore, the polynomial is:

= x² - (Sum of zeros)x + (Product of zeros)

= x² - (4 + 2)x + 4 × 2

= x² - 6x + 8

→ p(x) = x² - 6x + 8

★ Which is our required answer.

Answer:

• The required polynomial p(x) is x² - 6x + 8

Learn More:

1. Relationship between zeros and coefficients (Quadratic Polynomial)

Let f(x) = ax² + bx + c and let α and β be the zeros of f(x).

Therefore:

$\rm\longrightarrow\alpha+\beta=\dfrac{-b}{a}$

$\rm\longrightarrow\alpha\beta=\dfrac{c}{a}$

2. Relationship between zeros and coefficients (Cubic Polynomial)

Let f(x) = ax³ + bx² + cx + d and let α, β and γ be the zeros of f(x).

Therefore:

$\rm\longrightarrow \alpha+\beta+\gamma=\dfrac{-b}{a}$

$\rm\longrightarrow \alpha\beta+\beta\gamma+\alpha\gamma=\dfrac{c}{a}$

$\rm\longrightarrow \alpha\beta\gamma=\dfrac{-d}{a}$

Answer 2

(x - 4) is a factor

➻ p(4) = 0

⇒ 4 is also a Zero.

Quadratic Polynomial Format :

x² + (sum of zeros)x + (product of zeros)

⇒ x² + (4 + 2)x + (4 * 2)

⇒ x² + 6x + 8