Given that $x > 0$ and $x \neq 25$, fully simplify
$$
\frac{10 + 5x - 2x^{2} - x^{3}}{5 - \sqrt{x}}\n$$
Fully justify your answer. - AQA - A-Level Maths Pure - Question 6 - 2021 - Paper 3
Question 6
Given that $x > 0$ and $x \neq 25$, fully simplify
$$
\frac{10 + 5x - 2x^{2} - x^{3}}{5 - \sqrt{x}}\n$$
Fully justify your answer.
Worked Solution & Example Answer:Given that $x > 0$ and $x \neq 25$, fully simplify
$$
\frac{10 + 5x - 2x^{2} - x^{3}}{5 - \sqrt{x}}\n$$
Fully justify your answer. - AQA - A-Level Maths Pure - Question 6 - 2021 - Paper 3
Step 1
Identify the expression to simplify
96%
114 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
We start with the expression to simplify, which is:
5−x10+5x−2x2−x3.
Step 2
Factor the numerator
99%
104 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Next, we need to factor the numerator. Let's rearrange and resize terms:
10+5x−2x2−x3=−x3−2x2+5x+10.
Factoring the numerator gives:
−(x3+2x2−5x−10)=−(x+2)(x2−5).
Step 3
Identify common factors
96%
101 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Now, observe the denominator:
5−x.
By substituting back, we schedule that no common factor is directly observable. However, multiplying the denominator by its conjugate we reformulate expressing it as:
(5−x)(5+x)=25−x.
Step 4
Complete the simplification
98%
120 rated
Only available for registered users.
Sign up now to view full answer, or log in if you already have an account!
Answer
Finally, simplify the entire expression:
25−x−(x+2)(x2−5).
We note that cancellation can occur, yielding:
−(x+2).
Thus, the fully simplified form is:
−(x+2).
This representation is valid under the condition that x=25.
