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The polynomial $p(x)$ is given by $p(x) = x^3 + (b + 2)x^2 + 2(b + 2)x + 8$ where $b$ is a constant - AQA - A-Level Maths Pure - Question 11 - 2022 - Paper 1
Question 11
The polynomial $p(x)$ is given by $p(x) = x^3 + (b + 2)x^2 + 2(b + 2)x + 8$ where $b$ is a constant. 11 (a) Use the factor theorem to prove that $(x + 2)$ is ... show full transcript
Worked Solution & Example Answer:The polynomial $p(x)$ is given by $p(x) = x^3 + (b + 2)x^2 + 2(b + 2)x + 8$ where $b$ is a constant - AQA - A-Level Maths Pure - Question 11 - 2022 - Paper 1
Step 1
Use the factor theorem to prove that $(x + 2)$ is a factor of $p(x)$ for all values of $b$.
Answer
To use the factor theorem, we need to substitute into and show that it equals zero.
a. Substitute :
p(-2) &= (-2)^3 + (b + 2)(-2)^2 + 2(b + 2)(-2) + 8 \\ &= -8 + (b + 2)(4) - 4(b + 2) + 8 \\ &= -8 + 4b + 8 - 4b - 8 + 8 \\ &= 0. \end{aligned}$$ b. Since $p(-2) = 0$, by the factor theorem we conclude that $(x + 2)$ is a factor of $p(x)$ for all values of $b$.Step 2
Sketch a possible graph of $y = p(x)$ that meets the x-axis at exactly two points.
Answer
To meet the x-axis at exactly two points, the cubic graph must exhibit a single root and a repeated root. Hence, sketch a cubic graph with the following characteristics:
- A single root located at (the point of tangency).
- A y-intercept labeled at an appropriate value (for instance, when ).
- The graph can rise or fall at both ends, with an orientation that reflects a cubic function behavior.
- Ensure that only two points are touching the x-axis, at and another distinct point where the graph crosses the x-axis.
Step 3
Given $p(x)$ can be written as $p(x) = (x + 2)(x^2 + bx + 4)$, find the value of $b$.
Answer
To find the value of , we equate the quadratic factor from the expanded form of . First, we expand :
p(x) &= x(x^2 + bx + 4) + 2(x^2 + bx + 4) \\ &= x^3 + bx^2 + 4x + 2x^2 + 2bx + 8 \\ &= x^3 + (b + 2)x^2 + (4 + 2b)x + 8. \end{aligned}$$ By comparing coefficients with $p(x) = x^3 + (b + 2)x^2 + 2(b + 2)x + 8$, we analyze the quadratic term: From $2b + 4 = 2(b + 2)$, we simplify and find $b = 4$; substituting this confirms the intersection behavior with the x-axis. Thus, we conclude: $$b = 4.$$