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p(x) = 4x³ - 15x² - 48x - 36 - AQA - A-Level Maths Pure - Question 4 - 2020 - Paper 3
Question 4
p(x) = 4x³ - 15x² - 48x - 36. Use the factor theorem to prove that x - 6 is a factor of p(x). 4 (b) (i) Prove that the graph of y = p(x) intersects the x-axis at e... show full transcript
Worked Solution & Example Answer:p(x) = 4x³ - 15x² - 48x - 36 - AQA - A-Level Maths Pure - Question 4 - 2020 - Paper 3
Step 1
Use the factor theorem to prove that x - 6 is a factor of p(x).
Answer
To prove that x - 6 is a factor of p(x), we will use the factor theorem, which states that if f(c) = 0 for some polynomial f(x), then x - c is a factor of f(x).
Let p(x) = 4x³ - 15x² - 48x - 36. We will substitute x = 6 into p(x):
egin{align*} p(6) & = 4(6)^3 - 15(6)^2 - 48(6) - 36 \ & = 4(216) - 15(36) - 288 - 36 \ & = 864 - 540 - 288 - 36 \ & = 864 - 864 \ & = 0 \end{align*}Since p(6) = 0, we conclude that x - 6 is a factor of p(x).
Step 2
Prove that the graph of y = p(x) intersects the x-axis at exactly one point.
Answer
To prove that the graph of y = p(x) intersects the x-axis at exactly one point, we need to analyze the behavior of the function and its derivative.
- First, we find the derivative of p(x):
- Now, we will find the critical points by solving p'(x) = 0:
- Applying the quadratic formula:
Calculating the discriminant:
Thus, p(x) has two turning points. To confirm the intersection, we examine the sign changes of p(x). Since the leading coefficient is positive, the graph opens upwards, ensuring the existence of only one intersection with the x-axis.
Step 3