Question

find the value of p if 5x³+px²-x-3 and 3x³-4x²-3x+p have the same reminder when divided by (x-2)​

Answer 1

Given:-

• f(x) = 5x³ + px² - x - 3
• g(x) = 3x³ - 4x² - 3x + p
• The remainder on dividing f(x) and g(x) by (x - 2) is equal

To find:-

• The value of p

Answer:-

▪ According to remainder theorem, we know that if a polynomial p(x) is divided by (x - a), then the remainder is p(a).

▪For f(x):-

f(x) = 5x³ + px² - x - 3

On dividing by (x - 2),

→ Remainder = f(2) = (5 × 2³) + (p × 2²) - 2 - 3

→ Remainder = f(2) = (5 × 8) + (p × 4) - 5

→ Remainder = f(2) = 40 + 4p - 5

→ Remainder = f(2) = 35 + 4p

▪For g(x):-

g(x) = 3x³ - 4x² - 3x + p

On dividing by (x - 2),

→ Remainder = g(2) = (3 × 2³) - (4 × 2²) - (3 × 2) + p

→ Remainder = g(2) = (3 × 8) - (4 × 4) - 6 + p

→ Remainder = g(2) = 24 - 16 - 6 + p

→ Remainder = g(2) = 2 + p

▪ According to the question, the remainder on dividing f(x) and g(x) is equal.

So,

f(2) = g(2)

→ 35 + 4p = 2 + p

→ 4p - p = 2 - 35

→ 3p = -33

→ p = -33/3

→ p = -11 Ans.