If p(x) = 5(x + 1)(x - 2)(x - 4), at how many distinct points will the graph cut the x-axis?
Answer and explanation
Correct answer: Three
The x-axis intersections occur where p(x) = 0. Since 5 is non-zero, it cannot itself create a zero; the product is zero when at least one factor is zero. Thus x + 1 = 0 gives x = -1, x - 2 = 0 gives x = 2, and x - 4 = 0 gives x = 4. These are three different real values, so the graph has three distinct x-intercepts: (-1, 0), (2, 0), and (4, 0). Therefore option A is correct. The degree is three, so three distinct zeroes are possible and occur here. Option B or C would require repeated or missing roots, while option D wrongly counts the constant coefficient 5 as an additional zero.
Frequently asked questions
What is the correct answer to this question?
Three
Why is this the correct answer?
The x-axis intersections occur where p(x) = 0. Since 5 is non-zero, it cannot itself create a zero; the product is zero when at least one factor is zero. Thus x + 1 = 0 gives x = -1, x - 2 = 0 gives x = 2, and x - 4 = 0 gives x = 4. These are three different real values, so the graph has three distinct x-intercepts: (-1, 0), (2, 0), and (4, 0). Therefore option A is correct. The degree is three, so three distinct zeroes are possible and occur here. Option B or C would require repeated or missing roots, while option D wrongly counts the constant coefficient 5 as an additional zero.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Geometrical meaning of the zeroes of a polynomial..
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