If both x - 2 and x - 1/2 are factors of px² + 5x + r, show that p =… — Mathematics Part 1

Question 40 of 43advanced🔍 AnalyzeLong Answer5 marks

If both $x - 2$ and $x - \frac{1}{2}$ are factors of $px^2 + 5x + r$ , show that $p = r$ .

Correct Answer

p = r

Exercise: END-OF-CHAPTER EXERCISES | Q: 10 | (Page 23)

For More Understanding

Explanation

The solution applies the concept that a quadratic polynomial can be expressed as the product of its linear factors and a constant. By expanding the product of the given factors and comparing the coefficients of the resulting expression with the original polynomial—similar to comparing coefficients in identities like $x^2 + (a+b)x + ab$ shown in the textbook—we can determine the values of $p$ and $r$ , demonstrating that they are equal.

Solution Steps

Step 1: Since $(x - 2)$ and $\left(x - \frac{1}{2}\right)$ are factors of $px^2 + 5x + r$ , express the polynomial as:
$px^2 + 5x + r = k(x - 2)\left(x - \frac{1}{2}\right)$
Step 2: Expand the right side:
$k(x - 2)\left(x - \frac{1}{2}\right) = k\left[x^2 - \frac{1}{2}x - 2x + 1\right] = k\left[x^2 - \frac{5}{2}x + 1\right] = kx^2 - \frac{5k}{2}x + k$
Step 3: Compare the coefficients of corresponding terms on both sides.

For $x^2$ : $p = k$ .

For $x$ : $5 = -\frac{5k}{2}$ .

For the constant term: $r = k$ .
Step 4: Solve for $k$ from the $x$ coefficient equation:
$5 = -\frac{5k}{2}$
which gives $k = -2$ .
Step 5: Since $p = k$ and $r = k$ , it directly follows that $p = r$ (specifically, $p = r = -2$ ).

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