Question

. If sum of coefficients of terms of a polynomial is zero then ______is its factor

Answer 1

If sum of coefficients of terms of a polynomial is zero then (One) 1 is its factor

Solution :

A polynomial is a monomial or a sum of monomials

A monomial is number, a variable, or a product of numbers and variables with whole number exponents.

Degree of a polynomial is the highest degree of its monomials.

Factor Theorem. x – a is a factor of the polynomial p(x), if p(a) = 0.

Also, if x – a is a factor of p(x), then p(a) = 0,

where a is any real number.

This is an extension to remainder theorem where remainder is 0, i.e. p(a) = 0.

Polynomial of the form

P(x) = axⁿ + bxⁿ⁻¹ + ...  +  cx + d

if we substitute x = 1

P(1) = a(1)ⁿ + b(1)ⁿ⁻¹ + ...  +  c(1) + d

P(1) = a + b + ... + c + d

P(1) =  sum of coefficients of terms of the polynomial

But given that sum of coefficients of terms of   polynomial is zero

Hence P(1) = 0

so based on factor theorem 1 is the factor of the polynomial

Hence,  If sum of coefficients of terms of a polynomial is zero then (One) 1 is its factor

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