Question

in the equation $(x^2 - ax + 1)(x + 4) = x^3 + 2x^2 - 7x + 4$, a is a constant. if the equation is true for all values of x, what is the value of a?
○ 4
○ 2
○ -1
○ -2

Explanation:

Step1: Expand the left - hand side

We use the distributive property (FOIL method for polynomials) to expand \((x^{2}-Ax + 1)(x + 4)\).

$$
LATEXBLOCK0
$$

Step2: Compare coefficients

Since \((x^{2}-Ax + 1)(x + 4)=x^{3}+2x^{2}-7x + 4\) for all \(x\), the coefficients of corresponding powers of \(x\) on both sides must be equal.
For the \(x^{2}\) term: The coefficient of \(x^{2}\) on the left - hand side is \(4 - A\) and on the right - hand side is \(2\). So we have the equation \(4 - A=2\).
For the \(x\) term: The coefficient of \(x\) on the left - hand side is \(-4A + 1\) and on the right - hand side is \(-7\). Let's verify with the \(x^{2}\) term first.
From \(4 - A=2\), we can solve for \(A\) by subtracting \(4\) from both sides: \(-A=2 - 4=-2\), then multiplying both sides by \(- 1\) gives \(A = 2\).
We can also check with the \(x\) term. Substitute \(A = 2\) into \(-4A+1\): \(-4\times2 + 1=-8 + 1=-7\), which matches the coefficient of \(x\) on the right - hand side.

Answer:

The value of \(A\) is \(2\), so the option is 2.