Factor the trinomial:
$4 x^2-17 x+4$
To factor the quadratic trinomial \(4x^2 - 17x + 4\), we begin by identifying the coefficients. The trinomial is of the standard form \(ax^2 + bx + c\), where \(a = 4\), \(b = -17\), and \(c = 4\). Our goal is to express this trinomial as the product of two linear binomials of the form \((dx + e)(fx + g)\).
We use the AC method to factor the trinomial. First, we find the product of the leading coefficient \(a\) and the constant term \(c\), which is \(a \times c = 4 \times 4 = 16\). Next, we search for two numbers that multiply to \(16\) and add up to the middle coefficient \(b = -17\). Since the product is positive and the sum is negative, both numbers must be negative. Testing the factors of 16, we find that \(-1\) and \(-16\) satisfy both conditions: \((-1) \times (-16) = 16\) and \((-1) + (-16) = -17\).
Now, we rewrite the middle term \(-17x\) using the two numbers found: \(-16x\) and \(-1x\). The expression becomes:
\[4x^2 - 16x - 1x + 4\]
Next, we perform factoring by grouping. We group the first two terms and the last two terms:
\[(4x^2 - 16x) - (x - 4)\]
From the first group, we can factor out \(4x\):
\[4x(x - 4) - 1(x - 4)\]
Now, we see a common binomial factor \((x - 4)\). Factoring it out, we obtain:
\[(4x - 1)(x - 4)\]
We have successfully factored the trinomial \(4x^2 - 17x + 4\) into \((4x - 1)(x - 4)\). We can verify this by expanding the product:
\[(4x - 1)(x - 4) = 4x^2 - 16x - 1x + 4 = 4x^2 - 17x + 4\]
The expansion confirms our result. Thus, the factored form of the trinomial is \(\boxed{(4x - 1)(x - 4)}\).
Factoring quadratic expressions is a fundamental skill used in physics and engineering to solve problems involving motion. For example, if you are modeling the trajectory of a ball thrown into the air, its height \(h\) at time \(t\) might be given by a quadratic equation like \(h(t) = -4t^2 + 17t + 4\). Factoring the expression allows you to easily find the "roots" of the equation, which represent the times when the ball hits the ground (\(h=0\)). Mastering this process helps you solve for these critical points efficiently and understand the physical behavior of the object.