If \(-3\) is a root (zero) of a quadratic polynomial, which factor must the polynomial contain?
Answer and explanation
Correct answer: \(x+3\)
By the factor theorem, if \(r\) is a root of a polynomial then \(x-r\) is a factor. Here \(r=-3\), so the guaranteed factor is \(x-(-3)=x+3\). Option A (\(x-3\)) corresponds to root 3, not -3. Option C (\(3x-1\)) has root 1/3, unrelated to -3. Option D (\(x^2-9\)) equals \((x-3)(x+3)\); it would be a factor only if the other root were 3, so it is not guaranteed. Exam tip: for any root \(r\) use the linear factor \(x-r\); any nonzero constant multiple of that linear factor also vanishes at \(r\), but the canonical factor is \(x-r\).
Frequently asked questions
What is the correct answer to this question?
\(x+3\)
Why is this the correct answer?
By the factor theorem, if \(r\) is a root of a polynomial then \(x-r\) is a factor. Here \(r=-3\), so the guaranteed factor is \(x-(-3)=x+3\). Option A (\(x-3\)) corresponds to root 3, not -3. Option C (\(3x-1\)) has root 1/3, unrelated to -3. Option D (\(x^2-9\)) equals \((x-3)(x+3)\); it would be a factor only if the other root were 3, so it is not guaranteed. Exam tip: for any root \(r\) use the linear factor \(x-r\); any nonzero constant multiple of that linear factor also vanishes at \(r\), but the canonical factor is \(x-r\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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