If p(x) = x² + 7x + 3 is divided by x + 4, the remainder is

Solution:

Given, f(x) = x² + 7x + 3

The above function divided by (x + 4) implies that (x + 4) is a factor of the function.

The Remainder theorem states that when a polynomial f(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = f(k).

So, x = -4

f(-4) = (-4)² + 7(-4) + 3

f(-4) = 16 - 28 + 3

f(-4) = -9

Let us also try this using the long division method.

By division algorithm,

Dividend = Divisor × Quotient + Remainder

⇒ RHS = (x + 4) × (x + 3) - 9

= (x² + 3x + 4x + 12) - 9

= (x² + 7x + 12 ) - 9

= ( x² + 7x + 3) = LHS 

⇒ LHS = RHS 

Therefore, the remainder is -9.

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If p(x) = x² + 7x + 3 is divided by x + 4, the remainder is

Summary:

If p(x) = x² + 7x + 3 is divided by x + 4, the remainder is -9.