- Identify coefficients: \(a=16, b=0, c=-1\).
- Calculate \(ac = -16\) and recognize \(b=0\).
- Determine the two numbers that multiply to \(-16\) and add to \(0\), which are \(4\) and \(-4\).
- Factor the polynomial as \(\boxed{(4x - 1)(4x + 1)}\).
Explanation
- Identifying Coefficients and Calculating ac and b
To factor the polynomial \(16x^2 - 1\), we identify it in the standard quadratic form \(ax^2 + bx + c\). By comparing, we see that \(a = 16\), \(b = 0\), and \(c = -1\). The "X Method" (or factoring by grouping) requires us to find two numbers that multiply to \(ac\) and add to \(b\). First, we calculate \(ac = 16 \cdot (-1) = -16\). We know that \(b = 0\).
- Finding the Two Numbers for the X Method
We look for two numbers, \(m\) and \(n\), such that \(m \cdot n = -16\) and \(m + n = 0\). Examining the factors of \(-16\), we test \(4\) and \(-4\):
\[4 \cdot (-4) = -16\]
\[4 + (-4) = 0\]
Thus, the two numbers are \(4\) and \(-4\).
- Factoring the Polynomial
Using these numbers, we can rewrite the middle term of the polynomial as \(4x - 4x\):
\[16x^2 + 4x - 4x - 1\]
Now, group the terms:
\[(16x^2 + 4x) - (4x + 1)\]
Factor out the common terms from each group:
\[4x(4x + 1) - 1(4x + 1)\]
Finally, factor out the common binomial term \((4x + 1)\):
\[(4x - 1)(4x + 1)\]
This is also recognizable as a difference of squares \(A^2 - B^2 = (A-B)(A+B)\), where \(A = 4x\) and \(B = 1\), leading to the same result.
Examples
The difference of squares formula is a fundamental tool in algebra, useful for simplifying complex expressions and solving quadratic equations. For example, if you are designing a square garden of side \(4x\) and decide to remove a smaller square section of side \(1\) from the corner, the area remaining can be calculated as \((4x)^2 - 1^2\). Knowing how to factor this as \((4x-1)(4x+1)\) allows for quick mental math and efficient problem-solving in geometry and physics applications.