If \(p(x)=x^4+ax^2+b\) and \(p(0)=0\), which of the following conclusions is certainly true?
Answer and explanation
Correct answer: \(b=0\)
Substituting \(x=0\) gives \(p(0)=0^4+a\cdot0^2+b=b\). Hence the condition \(p(0)=0\) necessarily implies \(b=0\). It does not require \(a=0\); for example, \(a=1,b=0\) is possible. Also, \(p(1)=1+a\) is not always zero, and the polynomial generally has degree 4, not 2. Exam tip: evaluating a polynomial at zero isolates its constant term.
Frequently asked questions
What is the correct answer to this question?
\(b=0\)
Why is this the correct answer?
Substituting \(x=0\) gives \(p(0)=0^4+a\cdot0^2+b=b\). Hence the condition \(p(0)=0\) necessarily implies \(b=0\). It does not require \(a=0\); for example, \(a=1,b=0\) is possible. Also, \(p(1)=1+a\) is not always zero, and the polynomial generally has degree 4, not 2. Exam tip: evaluating a polynomial at zero isolates its constant term.
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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