Question

a polynomial has degree 4 and exactly one real zero at x = a with multiplicity 4. how many x-intercepts appear on its graph?
infinitely many intercepts along x = a
one intercept where the curve just touches
zero intercepts due to complex roots
two intercepts crossing twice
four intercepts crossing each time

Explanation:

Step1: Recall polynomial roots and intercepts

A polynomial's x - intercepts are the real zeros of the polynomial. For a polynomial of degree \( n \), the number of real and complex roots (counting multiplicities) is \( n \). Here, the polynomial is degree 4, and it has a real zero \( x = a \) with multiplicity 4. Complex roots come in conjugate pairs, but if all roots were complex, the degree would be even (since complex roots come in pairs), but here we have a real root with multiplicity 4. The x - intercepts are the real zeros. Since the only real zero is \( x=a \) (with multiplicity 4, which means the graph touches the x - axis at \( x = a \) and turns around, or just touches it, because the multiplicity is even), the number of x - intercepts is determined by the number of distinct real zeros. Since there's only one distinct real zero (\( x=a \)) with multiplicity 4, the graph will have one x - intercept where the curve just touches the x - axis at \( x = a \).

Step2: Analyze each option

• "Infinitely many intercepts along \( x = a \)": A polynomial of finite degree can't have infinitely many x - intercepts. So this is wrong.
• "One intercept where the curve just touches": Since the only real zero is \( x=a \) with multiplicity 4 (even multiplicity), the graph touches the x - axis at \( x = a \) and doesn't cross it (because even multiplicity means the graph bounces off or just touches the axis at that point). So this is correct.
• "Zero intercepts due to complex roots": But we have a real root \( x = a \), so there is at least one x - intercept. So this is wrong.
• "Two intercepts crossing twice": We have only one distinct real zero, so we can't have two x - intercepts. So this is wrong.
• "Four intercepts crossing each time": We have only one distinct real zero, so we can't have four x - intercepts. So this is wrong.

Answer:

One intercept where the curve just touches (the option with text "One intercept where the curve just touches")