If the polynomial \(p(x)=x^5+ax^3+b\) satisfies \(p(0)=0\), which of the following conclusions is certainly correct?
Answer and explanation
Correct answer: \(b=0\)
Substituting \(x=0\), we get \(p(0)=0^5+a\cdot0^3+b=b\). Hence the condition \(p(0)=0\) necessarily implies \(b=0\). The value of \(a\) is unrestricted; for example, \(a=2\) also satisfies the condition. Also, \(p(1)=1+a\), which is not zero for every possible value of \(a\). Since the coefficient of \(x^5\) is 1, the polynomial has degree 5, not 3. Exam tip: To find \(p(0)\), substitute zero; the result is the constant term.
Frequently asked questions
What is the correct answer to this question?
\(b=0\)
Why is this the correct answer?
Substituting \(x=0\), we get \(p(0)=0^5+a\cdot0^3+b=b\). Hence the condition \(p(0)=0\) necessarily implies \(b=0\). The value of \(a\) is unrestricted; for example, \(a=2\) also satisfies the condition. Also, \(p(1)=1+a\), which is not zero for every possible value of \(a\). Since the coefficient of \(x^5\) is 1, the polynomial has degree 5, not 3. Exam tip: To find \(p(0)\), substitute zero; the result is the 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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