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p(x) = 30x^3 - 7x^2 - 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) 12 (b) Factorise p(x) completely - AQA - A-Level Maths: Pure - Question 12 - 2018 - Paper 1
Question 12
p(x) = 30x^3 - 7x^2 - 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) 12 (b) Factorise p(x) completely. 12 (c) Prove that there are no real solutions to the... show full transcript
Worked Solution & Example Answer:p(x) = 30x^3 - 7x^2 - 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) 12 (b) Factorise p(x) completely - AQA - A-Level Maths: Pure - Question 12 - 2018 - Paper 1
Step 1
Prove that (2x + 1) is a factor of p(x)
Answer
To prove that (2x + 1) is a factor of p(x), we can use the Factor Theorem. According to the theorem, if (2x + 1) is a factor, then p(-1/2) should equal 0.
Let's evaluate p(-1/2):
egin{align*} p(-1/2) & = 30(-1/2)^3 - 7(-1/2)^2 - 7(-1/2) + 2 \\ & = 30(-1/8) - 7(1/4) + 3.5 + 2 \\ & = -3.75 - 1.75 + 3.5 + 2 \\ & = -3.75 - 1.75 + 5.5 \\ & = 0 e ight) ightarrow 0 ext{ (Hence, 2x + 1 is a factor)} \ ext{Thus, (2x + 1) is a factor of p(x).}Step 2
Factorise p(x) completely.
Answer
To factorise p(x) completely, we can use polynomial long division or synthetic division to divide p(x) by (2x + 1).
Starting with:
Dividing by (2x + 1), we find:
.
Next, we can further factor 15x^2 - 1 as a difference of squares:
.
Therefore, the complete factorization is:
.
Step 3
Prove that there are no real solutions to the equation 30 sec^2 x + 2 cos x = \frac{sec x + 1}{7}
Answer
To investigate the given equation for real solutions, rewrite it in terms of sine and cosine:
.
Multiply throughout by to eliminate the denominators:
This leads us to a cubic equation:
Upon analysis, examine the critical points and intervals for the function :
- is always positive for real values of , indicating there are no real roots.
Thus, we conclude that there are no real solutions to the equation.