If the sum of all coefficients of p(x)=2x³+kx²−8x+3 is 0, what is k?
Answer and explanation
Correct answer: 3
For a polynomial, the sum of all coefficients is obtained by substituting x=1, because every power of 1 equals 1. Thus p(1)=2(1)³+k(1)²−8(1)+3=2+k−8+3=k−3. The problem states that this sum is zero, so k−3=0 and therefore k=3. Hence option C is correct. Directly adding the coefficients gives the same equation: 2+k−8+3=0. Option A, B, or D would make the coefficient sum respectively −2, −1, or 1 rather than zero. The key idea is that the variable is not assigned an arbitrary value; x=1 is specifically used because it converts every coefficient term into that coefficient itself.
Frequently asked questions
What is the correct answer to this question?
3
Why is this the correct answer?
For a polynomial, the sum of all coefficients is obtained by substituting x=1, because every power of 1 equals 1. Thus p(1)=2(1)³+k(1)²−8(1)+3=2+k−8+3=k−3. The problem states that this sum is zero, so k−3=0 and therefore k=3. Hence option C is correct. Directly adding the coefficients gives the same equation: 2+k−8+3=0. Option A, B, or D would make the coefficient sum respectively −2, −1, or 1 rather than zero. The key idea is that the variable is not assigned an arbitrary value; x=1 is specifically used because it converts every coefficient term into that coefficient itself.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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